I think there is room to argue though, that distillation might violate copyright while training on large volumes of large material does not.
Nope. There is an argument about whether training a model on source material violates copyright or not. Current court decisions in the US say that the training does not, but you must have acquired the material legally in the first place.
Current court decisions in the US say that the raw output of a model is not copyrightable. Never mind that the people doing the distilling have paid for that output so even if it were copyrightable they'd own it.
The only thing happening is violation of terms of service which haven't been tested in court and would (hopefully) fail that test. Otherwise good luck with any software you use to produce anything, compilers included. And that's why these companies are directly lobbying the US government to do, uh, something, about it. They're trying to turn a bug in their business model into a geopolitical issue.