Hiring companies certainly don’t want bots to write job applications. They are already busy weeding out the AI-written applications and bots would only accelerate their problem. Hiring companies happen to be paying customers of LinkedIn.
Job applications aren't the only use case for using LinkedIn in this connected way, but even on that topic -- I think we are moving pretty quickly to no longer need to "weed out" AI-written applications.
As adoption increases, there's going to be a whole spectrum of AI-enabled work that you see out there. So something that doesn't appear to be AI written is not necessarily pure & free of AI. Not to mention the models themselves getting better at not sounding AI-style canned. If you want to have a filter for lazy applications that are written with a 10-word prompt using 4o, sure, that is actually pretty trivial to do with OpenAI's own models, but is there another reason you think companies "don't want bots to write job applications"?