Minor addition: This question reminded me of my reply in this thread.
It is hard to beat the simplicity of Dr. Harrell’s recommendation of semi-parametric modelling as a general rule. I’m pretty surprised at how much of applied stats can be compressed into proportional odds models. This strikes me as something only someone with decades of applied stats experience would figure out.
If there is a strong reason to prefer a frequentist parametric approach, I’d look at the adaptive methods of O’Gorman before considering textbook robust methods (despite their theoretical and historical importance).
Addendum: Links to O’Gorman’s texts are broken and not easily available from archive.org. This is an early paper on adaptive regression worth study.
O’Gorman, T.W. (2002), An adaptive test of significance for a subset of regression coefficients. Statist. Med., 21: 3527-3542. https://doi.org/10.1002/sim.1278