Why report test statistics in a publication? Some style guides instruct authors to report not only the results of a hypothesis test, but also the value of the test statistic from which it was calculated. For example, APA Style suggests t(DOF)=t statistic, p=p value for reporting the results of a t-test, as shown in the following examples, taken from here¹.

One sample: “Younger teens woke up earlier (M = 7:30, SD = .45) than teens in general, t(33) = 2.10, p = 0.31″
Dependent/Independent samples: “Younger teens indicated a significant preference for video games (M = 7.45, SD = 2.51) than books (M = 4.22, SD = 2.23), t(15) = 4.00, p < .001.”

The $p$-value is obviously useful², as it tells you if the results are unlikely under the null hypothesis. Descriptive statistics are important for understanding the characteristics of the data/subject pool, as well as the size of any purported effect.
The $t$-statistic (or similar for $\chi^2$ tests) seems more like an intermediate step, needed to get one from the other. Likewise the degree(s) of freedom are often closely related to the number of data points, but the actual value of $N$ seems easier to interpret.
How does including the test statistic help me interpret these results? Is it merely convention, an aid to meta-analysis, or can astute reader learn something from these numbers?
I'm particularly interested in the case of "simple" tests; I can imagine that $F(x,y)$ tells you something about the design of an ANOVA that might not be clear from a written description.



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*These examples don't actually seem very good to me; the mean of the within-subject differences would be more informative, for example.


*In a NHST framework, of course. Assume we're happy working in it for the purposes of this question.
 A: I'm certainly not a mind reader as to why or how these recommendations came into being; however, I can at least speculate and share some personal experience in APA.
As you observed in your response to @BruceET, the APA guidelines for reporting the test statistic do predate the major position papers on the misuse of p-values. As such, the recommendations also predate the transition to assuming that effect sizes will be reported for every test. Prior to such a requirement, interested readers could at least compute some rough effect measures from the test statistics (e.g., d = t/sqrt(df)).
Ultimately, though, I think the short answer is as a matter of transparency. If someone reports their results as t(30) = 1.50, p < .001, d = 1.25, then readers (ideally this would be caught by the reviewers/editors) can look at those values and see clearly that this is incorrect. Similarly, if someone were to just report, "The means differed significantly, p < .001," then we may be missing some vital information. I think your question is fair in a world where research know exactly what they're doing and are making well-informed decisions; however, the reality is that statistical software packages can't always warn someone that they are requesting a non-sensical test. I think this also matters for questions of parametric versus non-parametric tests. Saying something like, "the variables were significantly correlated, p < .05" doesn't tell us the kind of correlation and thus might not help us understand the potential meaning of that result.
On the note of degrees of freedom, I do think part of that is related to ways of computing effect sizes, but I think it is also a transparency issue as well. Just because the total sample for a study may be large, once you factor in missing observations, the n for specific statistical tests and the N of the sample can be fairly different. Should an author find that their test was non-significant in the entire sample, they may be tempted to start looking at subgroups or portions of the sample, and reporting the degrees of freedom for each of those results helps prevent mis-interpretation.
In short, I think in a perfect world where everyone is very responsible in their research and everyone understand the statistics they are using, there wouldn't be any need for reporting the statistics themselves. I think, however, as a matter of pragmatism and good faith effort for transparency in research, reference styles like APA recommend such practices.
