Should we use the NHST framework when we want to accept the null hypothesis? Given that what a p-value tells us is the probability of observing the data assuming the null is true, should we ever use null hypothesis significance testing (NHST) when we would rather accept the null hypothesis?
Take the Shapiro-Wilk test through which we might want to establish the normality of our data. Given a p-value of .36 and a conventional $\alpha$ of .05, we would fail to reject the null hypothesis. And the typical follow up is the suggestion that our data are normal.
However, the specific interpretation of $p=.36$ is: assuming our data are normal, the probability of observing these data is 36%. Considering this, it does not seem like good practice to act like failure to reject the null is equivalent to accepting the null. Am I correct?
Absence of evidence is not evidence of absence. Why then is the NHST framework often used when we would want to accept the null?
 A: You are right that a lack of (sufficient) evidence of a deviation from the null hypothesis is not necessarily  evidence for a lack of a deviation. As already noted one fails to reject the null hypothesis instead of accepting it. In fact the procedure you describe (use of test with normality assumption or rank test depending on a test of normality of regression residuals) is known to lead to tests that do not have the nominal significance level.
The standard solution is to make the null hypothesis the alternative - although that can be difficult. E.g. for the point null treatment A compared to treatment B has no effect on blood pressure (versus the alternative that there is a difference), an "inverted" alternative hypothesis for showing that the treatments are equivalent might be "the absolute difference in blood pressure between treatments A and B is at most 5 mmHg" (with the null being that the difference is either smaller than -5 or greater than 5). To come back yo you example, an irrelevant deviation from normality within some broad general class of distributions is a lot harder to define. 
A: First, the p-value is not 

the probability of observing the data assuming the null is true,

it is the probability of getting a test statistic at least as extreme as the one we got, given that the null is true.
Second, for normality, there really aren't good tests for using the NHST, regardless of what we want to do. It's much better to use graphical methods. Or, if the reason you are testing normality is to see whether some statistical model is appropriate, then you can do the version that assumes normality and one that doesn't and compare results.
Third, more generally, there are tests of equivalence, tests of superiority and tests of noninferiority. 
A: Yes, NHST can confirm the null hypothesis. An almost sure hypothesis test, using a significance level of $n^{-p}$ for $p>1$, will accept the null whenever it is true and reject the null whenever it is false in any sufficiently large sample with probability one. 
