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.