Whenever AI is mentioned lots of people in the Linux space immediately react negatively. Creators like TheLinuxExperiment on YouTube always feel the need to add a disclaimer that "some people think AI is problematic" or something along those lines if an AI topic is discussed. I get that AI has many problems but at the same time the potential it has is immense, especially as an assistant on personal computers (just look at what "Apple Intelligence" seems to be capable of.) Gnome and other desktops need to start working on integrating FOSS AI models so that we don't become obsolete. Using an AI-less desktop may be akin to hand copying books after the printing press revolution. If you think of specific problems it is better to point them out and try think of solutions, not reject the technology as a whole.
TLDR: A lot of ludite sentiments around AI in Linux community.
In addition to everything everyone else has already said, why does this have anything to do with desktop environments at all? Remember, most open-source software comes from one or two individual programmers scratching a personal itch—not all of it is part of your DE, nor should it be. If someone writes an open-source LLM-driven program that does something useful to a significant segment of the Linux community, it will get packaged by at least some distros, accrete various front-ends in different toolkits, and so on.
However, I don't think that day is coming soon. Most of the things "Apple Intelligence" seems to be intended to fuel are either useless or downright offputting to me, and I doubt I'm the only one—for instance, I don't talk to my computer unless I'm cussing it out, and I'd rather it not understand that. My guess is that the first desktop-directed offering we see in Linux is going to be an image generator frontend, which I don't need but can see use cases for even if usage of the generated images is restricted (see below).
Anyway, if this is your particular itch, you can scratch it—by paying someone to write the code for you (or starting a crowdfunding campaign for same), if you don't know how to do it yourself. If this isn't worth money or time to you, why should it be to anyone else? Linux isn't in competition with the proprietary OSs in the way you seem to think.
As for why LLMs are so heavily disliked in the open-source community? There are three reasons:
Item 1 can theoretically be solved by bigger and better AI models, but 2 and 3 can't be. They have to be decided by the courts, and at an international level, too. We might even be talking treaty negotiations. I'd be surprised if that takes less than ten years. In the meanwhile, for instance, it's very, very dangerous for any open-source project to accept a code patch written with the aid of an LLM—depending on the conclusion the courts come to, it might have to be torn out down the line, along with everything built on top of it. The inability to use LLM output for open source or commercial purposes without taking a big legal risk kneecaps the value of the applications. Unlike Apple or Microsoft, the Linux community can't bribe enough judges to make the problems disappear.