I fail to see how that follows from what Sander wrote. The quote from Ernst in the Permutation Methods paper (who was also a co-author of the initial paper I linked to) described the problem: parametric assumptions that require large sample properties to hold for our finite, often very small sample, but we don’t know how quickly we converge to the limit.
Philip Good in his book Permutation, Parametric, and Boostrap Tests of Hypotheses (3rd ed, p. 153-154) gave a real world example (categorical data examined via Chi-square statistics) where the permutation p values and the large sample approximations differed by a factor of 10! The permutation test detected differences that asymptotic approximations did not.
I think it is unfortunate permutation methods are invariably perceived as tests; interval estimates and entire compatibility distributions (more commonly called “confidence” distributions) can be created from them. In the case of RCTs they substitute an unverifiable assumption with a design based proposition that can be taken as correct by construction (if you accept the research report as described).
Related article that @ESMD might appreciate:
https://escholarship.org/uc/item/6hb3k0nz