The objection to CSAM is rooted in how it is (inhumanely) produced; people are not merely objecting to a GET request.

Yes, they're objecting to people training on data they don't have the right to, not just the GET request as you suggest.

If you distribute child porn, that is a crime. But if you crawl every image on the web and then train a model that can then synthesize child porn, the current legal model apparently has no concept of this and it is treated completely differently.

Generally, I am more interested in how this effects copyright. These AI companies just have free reign to convert copyrighted works into the public domain through the proxy of over-trained AI models. If you release something as GPL, they can strip the license, but the same is not true of closed-source code which isn't trained on.

Indeed, and neither is that what people are objecting to with regard to AI training data.