I believe that a further restriction of fair use to what is produced by a generative AI would also give a lot of tools for large copyright holders (e.g. Disney) for suing and removing fan art for anything that "is in the style of" or "makes use of the same characters" including uses that would previously be considered beneath their notice and thus not worth their time (the amount the lawyers would cost combined with the difficulty in doing it).

The lawyers won't "mess it up" but rather that the large copyright holders will be able to use this to enforce these restrictions on human generated fan art just as easily as they (and other humans) will be able to enforce it on generative AI.

While it won't be possible to excise the idea of "draw like Disney" from someone's brain, it will be possible to sue the creator to remove the work.