I'm not a statistician, but I'll try to weigh-in.
Note: edited somewhat in response to comments by OP, to make this response more relevant for the linear regression case.
For the p value to be valid, the assumptions for the test need to be met. These may include normality of errors for some tests, but other tests will have different assumptions. That is, if alternatives to classic linear regression are considered, the assumptions of p values for parameters for those models may not include that assumption. Questions about what assumptions or complications go into considering the reliability of robust models depend upon the specific robust model being considered.
Relying on the central limit theorem to ensure the normality of parameters without examining the data is not a great practice. When thinking about e.g. if the sample mean is likely to come from a normal distribution, if the population distribution is very skewed, it may take a very large sample size to ensure an approximately normal distribution of means (and 30 is not necessarily a large sample size). (See Rand Wilcox quote below which considers specifically the two sample test, but I think is instructive.) It's not entirely clear to me how these considerations can be used practically in assessing the appropriateness of classic linear regression.
Rand Wilcox, 2017, Modern Statistics for the Social and Behavioral Sciences. Section 7.3.4.
Three Modern Insights Regarding Methods for Comparing Means
There have been three modern insights regarding methods for comparing
means, each of which has already been described. But these insights
are of such fundamental importance that it is worth summarizing them
• Resorting to the central limit theorem in order to justify the
normality assumption can be highly unsatisfactory when working with
means. Under general conditions, hundreds of observations might be
needed to get reasonably accurate confidence intervals and good
control over the probability of a Type I error. Or in the context of
Tukey's three-decision rule, hundreds of observations might be needed
to be reasonably certain which group has the largest mean. When using
Student's T, rather than Welch's test, concerns arise regardless of
how large the sample sizes might be.
• Practical concerns about heteroscedasticity (unequal variances) have
been found to much more serious than once thought. All indications are
that it is generally better to use a method that allows unequal
• When comparing means, power can be very low relative to other
methods that might be used. Both differences in skewness and outliers
can result in relatively low power. Even if no outliers are found,
differences in skewness might create practical problems. Certainly
there are exceptions. But all indications are that it is prudent not
to assume that these concerns can be ignored.
Despite the negative features just listed, there is one positive
feature of Student's T is worth stressing. If the groups being
compared do not differ in any manner, meaning that they have identical
distributions, so in particular the groups have equal means, equal
variances, and the same amount of skewness, Student's T appears to
control the probability of a Type I error reasonably well under
nonnormality. That is, when Student's T rejects, it is reasonable to
conclude that the groups differ in some manner, but the nature of the
difference, or the main reason Student's T rejected, is unclear. Also
note that from the point of view of Tukey's three-decision rule,
testing and rejecting the hypothesis of identical distributions is not