This is a great advantage of the AGPL IMO. You can release your free software/open source code, and big tech companies won't use your code and make lots of money.
Some companies tried to relicence their code to some 'source available'/'non commerical' licence because they didn't like the MIT's lack of conditions. AGPL could have helped.
As an example, let's say that I have a game. That game happens to want to have a way of rendering some HTML content for printing / making a PDF. It's OK to ship an open source thing with your game, like a hypothetical FrozenFluff web-browser licensed in AGPL. There are two distinct entities there: a FF binary that gets called with a file to render (which is AGPL and an AGPL processed result that is under the copyright of whomever's material was rendered) and another distinct binary the GameX that happened to call an external process.
The line to cross there is that the AGPL stuff can't be linked against your own program. I'm even more unsure how a module might or might not be licensed. I think since that's an addin then the code for the module (including headers) would have to be AGPL compatible and released, but that the rest could probably be closed.
This line is less clear when it comes to websites however. I think the best equivalent would be to make a wrapper interface that is AGPL licensed, published, and links to/calls the AGPL software. It should have a distinct URL. That URL might only be on the localhost (for security) and would hand back whatever results exist. The proprietary interface could talk to that URL and get results.
I am also not a lawyer, however my take on many legal things is that it doesn't matter what my take is, or even if courts will agree with my take, what matters is whether the benefit is worth the hassle if I end up needing a lawyer to deal with it.
Even if you "win," the engagement with lawyers can cost you so much that it becomes a pyrrhic victory, and the amount of time required to deal with certain legal squabbles is astounding.
So sometimes, the right thing to do is to shy clear of anything that carries with it some risk of legal potholes.
Honestly they are probably correct to be worried about this because it's pretty unclear whether or not it would be covered. At the very least, by making it against the TOS it is clear that it's the users fault not theirs if there is a violation.