Objective criteria for assumption violations that do not utilize p-values? Suppose we have a standard regression model and want to identify whether we have violated the assumptions of the model. Traditionally, we might utilize a significance test to determine whether (for example) the residuals are normally distributed (where the null model states that the residuals belong to a normal distribution).
I think there's consensus that using hypothesis tests to evaluate model assumptions isn't the best idea. These tests are sensitive to sample size (failing to detect major departures when N is small, and detecting even minor discrepancies when N is large). Personally, I think the biggest problem with these tests is they evaluate whether violated assumptions can be detected, not whether they are problematic.
When I evaluate assumptions, I exclusively use visuals. Unfortunately, visuals are more subjective (and they don't really tell you whether the observed violation is problematic). So, here's my question:
Are there any objective ways to evaluate statistical assumptions that do not rely on significance tests?
And, do these objective means of evaluating assumption evaluate whether the assumption violation matters, rather than whether it can be detected?
I'd love to see references to articles that I can study, if you have it. Thanks!
Edit
Some of the comments/answers mention that it depends on what one's research goals are. Yes, I absolutely agree and appreciate you all pointing this out. I suppose what I'm asking for is some literature that says, "If your goal is A, index B is appropriate. If your goal is C, criteria D is appropriate."

This is not a complete answer, and it's not exactly "objective", but a useful tool is what is known as posterior predictive checks in a Bayesian context (but probably by other names elsewhere). In this, you simulate data from the distribution implied by your model, and compare it to your actual data, usually by plotting them side-by-side in whatever fashion makes sense for your problem. Mismatches then suggest assumptions that don't hold - e.g. your simulated data is symmetrical where the real data is skewed, or the real data has fatter tails.
