Are effect sizes really superior to p-values? Lots of emphasis is placed on relying on and reporting effect sizes rather than p-values in applied research (e.g. quotes further below). 
But is it not the case that an effect size just like a p-value is a random variable and as such can vary from sample to sample when the same experiment is repeated? In other words, I'm asking what statistical features (e.g., effect size is less variable from sample to sample than p-value) make effect sizes better evidence-measuring indices than p-values?
I should, however, mention an important fact that separates a p-value from an effect size. That is, an effect size is something to be estimated because it has a population parameter but a p-value is nothing to be estimated because it doesn't have any population parameter.  
To me, effect size is simply a metric that in certain areas of research (e.g., human research) helps transforming empirical findings that come from various researcher-developed measurement tools into a common metric (fair to say using this metric human research can better fit the quant research club).
Maybe if we take a simple proportion as an effect size, the following (in R) is what shows the supremacy of effect sizes over p-values? (p-value changes but effect size doesn't)
binom.test(55, 100, .5)  ## p-value = 0.3682  ## proportion of success 55% 

binom.test(550, 1000, .5) ## p-value = 0.001731 ## proportion of success 55%

Note that most effect sizes are linearly related to a test statistic. Thus, it is an easy step to do null-hypothesis testing using effect sizes. 
For example, t statistic resulted from a pre-post design can easily be converted to a corresponding Cohen's d effect size. As such, distribution of Cohen's d is simply the scale-location version of a t distribution.
The quotes:

Because p-values are confounded indices, in theory 100 studies with
  varying sample sizes and 100 different effect sizes could each have
  the same single p-value, and 100 studies with the same single effect
  size could each have 100 different values for p-value.

or

p-value is a random variable that varies from sample to sample. . . .
  Consequently, it is not appropriate to compare the p-values from two
  distinct experiments, or from tests on two variables measured in the
  same experiment, and declare that one is more significant than the
  other?

Citations:
Thompson, B. (2006). Foundations of behavioral statistics: An
insight-based approach. New York, NY: Guilford Press.
Good, P. I., & Hardin, J. W. (2003). Common errors in statistics (and how to avoid them). New York: Wiley.
 A: I currently work in the data science field, and before then I worked in education research. While at each "career" I've collaborated with people who did not come from a formal background in statistics, and where emphasis of statistical (and practical) significance is heavily placed on the p-value. I've learned include and emphasize effect sizes in my analyses because there is a difference between statistical significance and practical significance. 
Generally, the people I worked with cared about one thing "does our program/feature make and impact, yes or no?". To a question like this, you can do something as simple as a t-test and report to them "yes, your program/feature makes a difference". But how large or small is this "difference"?
First, before I begin delving into this topic, I'd like to summarize what we refer to when speaking of effect sizes

Effect size is simply a way of quantifying the size of the difference between two groups. [...] It is particularly valuable for quantifying the effectiveness of a particular intervention, relative to some comparison. It allows us to move beyond the simplistic, 'Does it work or not?' to the far more sophisticated, 'How well does it work in a range of contexts?' Moreover, by placing the emphasis on the most important aspect of an intervention - the size of the effect - rather than its statistical significance (which conflates effect size and sample size), it promotes a more scientific approach to the accumulation of knowledge. For these reasons, effect size is an important tool in reporting and interpreting effectiveness.

It's the Effect Size, Stupid: What effect size is and why it is important
Next, what is a p-value, and what information does it provide us? Well, a p-value, in as few words as possible, is a probability that the observed difference from the null distribution is by pure chance. We therefore reject (or fail to accept) the null hypothesis when this p-value is smaller than a threshold ($\alpha$). 

Why Isn't the P Value Enough?
Statistical significance is the probability that the observed difference between two groups is due to chance. If the P value is larger than the alpha level chosen (eg, .05), any observed difference is assumed to be explained by sampling variability. With a sufficiently large sample, a statistical test will almost always demonstrate a significant difference, unless there is no effect whatsoever, that is, when the effect size is exactly zero; yet very small differences, even if significant, are often meaningless. Thus, reporting only the significant P value for an analysis is not adequate for readers to fully understand the results.

And to corroborate @DarrenJames's comments regarding large sample sizes

For example, if a sample size is 10 000, a significant P value is likely to be found even when the difference in outcomes between groups is negligible and may not justify an expensive or time-consuming intervention over another. The level of significance by itself does not predict effect size. Unlike significance tests, effect size is independent of sample size. Statistical significance, on the other hand, depends upon both sample size and effect size. For this reason, P values are considered to be confounded because of their dependence on sample size. Sometimes a statistically significant result means only that a huge sample size was used. [There is a mistaken view that this behaviour represents a bias against the null hypothesis. Why does frequentist hypothesis testing become biased towards rejecting the null hypothesis with sufficiently large samples? ]

Using Effect Size—or Why the P Value Is Not Enough
Report Both P-value and Effect Sizes
Now to answer the question, are effect sizes superior to p-values? I would argue, that these each serve as importance components in statistical analysis that cannot be compared in such terms, and should be reported together. The p-value is a statistic to indicate statistical significance (difference from the null distribution), where the effect size puts into words how much of a difference there is.
As an example, say your supervisor, Bob, who is not very stats-friendly is interested in seeing if there was a significant relationship between wt (weight) and mpg (miles per gallon). You start the analysis with hypotheses
$$
H_0: \beta_{mpg} = 0 \text{ vs } H_A: \beta_{mpg} \neq 0
$$
being tested at $\alpha = 0.05$
> data("mtcars")
> 
> fit = lm(formula = mpg ~ wt, data = mtcars)
> 
> summary(fit)

Call:
lm(formula = mpg ~ wt, data = mtcars)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.5432 -2.3647 -0.1252  1.4096  6.8727 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
wt           -5.3445     0.5591  -9.559 1.29e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.046 on 30 degrees of freedom
Multiple R-squared:  0.7528,    Adjusted R-squared:  0.7446 
F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

From the summary output we can see that we have a t-statistic with a very small p-value. We can comfortably reject the null hypothesis and report that $\beta_{mpg} \neq 0$. However, your boss asks, well, how different is it? You can tell Bob, "well, it looks like there is a negative linear relationship between mpg and wt. Also, can be summarized that for every increased unit in wt there is a decrease of 5.3445 in mpg" 
Thus, you were able to conclude that results were statistically significant, and communicate the significance in practical terms.
I hope this was useful in answering your question.
A: From the perspective of an Epidemiologist, on why I prefer effect sizes over p-values (though as some people have noted, it's something of a false dichotomy):


*

*The effect size tells me what I actually want, the p-value just tells me if it's distinguishable from null. A relative risk of 1.0001, 1.5, 5, and 50 might all have the same p-value associated with them, but mean vastly different things in terms of what we might need to do at a population level.

*Relying on a p-value reinforces the notion that significance-based hypothesis testing is the end-all, be-all of evidence. Consider the following two statements: "Doctors smiling at patients was not significantly associated with an adverse outcome during their hospital stay." vs. "Patients who had their doctor smile at them were 50% less likely to have an adverse outcome (p = 0.086)." Would you still, maybe, given it has absolutely no cost, consider suggesting doctors smile at their patients?

*I work with a lot of stochastic simulation models, wherein sample size is a function of computing power and patience, and p-values are essentially meaningless. I have managed to get p < 0.05 results for things that have absolutely no clinical or public health relevance.

A: The advice to provide effect sizes rather than P-values is based on a false dichotomy and is silly. Why not present both?
Scientific conclusions should be based on a rational assessment of available evidence and theory. P-values and observed effect sizes alone or together are not enough.
Neither of the quoted passages that you supply is helpful. Of course P-values vary from experiment to experiment, the strength of evidence in the data varies from experiment to experiment. The P-value is just a numerical extraction of that evidence by way of the statistical model. Given the nature of the P-value, it is very rarely relevant to analytical purposes to compare one P-value with another, so perhaps that is what the quote author is trying to convey. 
If you find yourself wanting to compare P-values then you probably should have performed a significance test on a different arrangement of the data in order to sensibly answer the question of interest. See these questions:
p-values for p-values? and 
If one group's mean differs from zero but the other does not, can we conclude that the groups are different?
So, the answer to your question is complex. I do not find dichotomous responses to data based on either P-values or effect sizes to be useful, so are effect sizes superior to P-values? Yes, no, sometimes, maybe, and it depends on your purpose.
A: In the context of applied research, effect sizes are necessary for readers to interpret the practical significance (as opposed to statistical significance) of the findings.  In general, p-values are far more sensitive to sample size than effect sizes are.  If an experiment measures an effect size accurately (i.e. it is sufficiently close to the population parameter it is estimating) but yields a non-significant p-value then, all things being equal, increasing the sample size will result in the same effect size but a lower p-value.  This can be demonstrated with power analyses or simulations.  
In light of this, it is possible to achieve highly significant p-values for effect sizes that have no practical significance.  In contrast, study designs with low power can produce non-significant p-values for effect sizes of great practical importance. 
It is difficult to discuss the concepts of statistical significance vis-a-vis effect size without a specific real-world application. As an example, consider an experiment that evaluates the effect of a new studying method on students' grade point average (GPA).  I would argue that an effect size of 0.01 grade points has little practical significance (i.e. 2.50 compared to 2.51).  Assuming a sample size of 2,000 students in both treatment and control groups, and a population standard deviation of 0.5 grade points:
set.seed(12345)
control.data <- rnorm(n=2000, mean = 2.5, sd = 0.5)
set.seed(12345)
treatment.data <- rnorm(n=2000, mean = 2.51, sd = 0.5)
t.test(x = control.data, y = treatment.data, alternative = "two.sided", var.equal = TRUE) 

treatment sample mean = 2.51
control sample mean = 2.50
effect size = 2.51  - 2.50  = 0.01
p = 0.53
Increasing the sample size to 20,000 students and holding everything else constant yields a significant p-value:
set.seed(12345)
control.data <- rnorm(n=20000, mean = 2.5, sd = 0.5)
set.seed(12345)
treatment.data <- rnorm(n=20000, mean = 2.51, sd = 0.5)
t.test(x = control.data, y = treatment.data, alternative = "two.sided", var.equal = TRUE)  

treatment sample mean = 2.51
control sample mean = 2.50
effect size = 2.51  - 2.50  = 0.01
p = 0.044
Obviously it's no trivial thing to increase the sample size by an order of magnitude!  However, I think we can all agree that the practical improvement offered by this study method is negligible.  If we relied solely on the p-value then we might believe otherwise in the n=20,000 case.  
Personally I advocate for reporting both p-values and effect sizes.  And bonus points for t- or F-statistics, degrees of freedom and model diagnostics!
