"A property holds if it cannot be rejected" Most examples in my Time Series Analysis slides and Multivariate Analysis textbook (Applied Multivariate Statistical Analysis, 6th Edition) conduct a hypothesis testing on data's normality, or zero correlation, or independence, or homogeneity of variance, or whatever, and readily assume these properties to be true on failure to reject the hypothesis. Actually, I am explicitly taught that a property holds if it cannot be rejected.
For example, in Example 6.14 (A profile analysis of love and marriage data) from the textbook AMSA, authors "conclude that the hypothesis of parallel profiles for men and women is tenable" when $H_0$ cannot be rejected. In addition, they also said "... we cannot reject the hypothesis that the profiles are coincident. That is, the responses of men and women to the four questions posed appear to be the same." Here is the whole example: P.325, P.326, P.327, P.328, where the quotes are from the bottom of P.327 and the top of P.328 respectively.
This seems ridiculous, as absence of evidence is not evidence of an absence. Even if $p = 1$, it's still circular reasoning, let along cases where $p$ isn't even remotely close to 1. Back when I was learning Mathematical Statistics, the professor would kick their ass if someone accept $H_0$ on failure to reject it, but the standard is clearly loosened a lot in upper division courses.
I just want to know is this practice a de facto procedure in intermediate Statistic courses like Time Series Analysis and Multivariate Analysis? If so, what the rationale?
 A: Here is a demonstration using four samples of size $n = 15$ from uniform, gamma, beta, and normal populations.
Tests of normality. In all four cases the data 'pass' a Shapiro-Wilk test
for normality. That is, we cannot reject the null hypothesis that data are normal. Approximate P-values for the normality test are 0.40, 0.48, 0.09, and 0.36, respectively--all clearly above 0.05. 
The PDFs of the first three distributions are all substantially different from normal, but $n = 15$ observations are not enough to
detect the departure from normality.
Robustness of the t test: Also, in all four cases we use a one-sample t test
to test a null hypothesis about the population mean
that is false. For all four samples, the null hypothesis is (correctly) rejected. Even for a sample size as
small as $n = 15,$ the test detects that the various hypothetical values of the population mean  are not correct. 
The t test is said to be 'robust' against
departures from normality---as long as there are no
extreme outliers. Although the first three samples are non-normal, the t test still gives useful results.
Here are boxplots of the four samples;

set.seed(1234);  n = 15
u = runif(n);  shapiro.test(u)$p.val;  t.test(u, mu=.7)$p.val
[1] 0.3963761   # P-value for normality test
 [1] 0.0145073  # P-value for t test

v = rgamma(n, 5, 1); shapiro.test(v)$p.val; 
t.test(v, mu=2)$p.val
[1] 0.4827358
 [1] 5.112448e-05

w = rbeta(n,3,4); shapiro.test(w)$p.val; 
t.test(w,mu=.7)$p.val
[1] 0.08774671
 [1] 9.815778e-05

z = rnorm(n); shapiro.test(z)$p.val; 
t.test(z,mu=.7)$p.val
[1] 0.3592276
 [1] 0.003356458

Note: The seed for the simulation is given so that you can reproduce exactly the same simulated data used here. Different seeds might give substantially different results.
