Part of it is the guilt-by-association with the other bad writing habits of LLMs, but I think a lot of it is just that LLMs genuinely overuse them, and that homogeneity is grating just like it's grating when you notice a text reuses a particular noticeable word or whatever. As a fellow em-dash user, I have sometimes noticed myself overusing them too, and revised accordingly, starting well before the proliferation of this particular cancer.

So I think you can keep using em-dashes without being associated with LLMs as long as you reserve them for particularly effective/tasteful occasions.