The OP seems to think we should approach cover letters with the same care as if we were writing the great American novel. I disagree. They are meant to be somewhat canned, but a good one will efficiently direct the reviewers attention to the parts of your resume which are most relevant to the job and give the impression that you are specifically interested in the role. This is a task generative AI can accomplish with appropriate direction.

Obviously a generic prompt is going to get a generic response, but if you put in the work to create a prompt with adequate context and descriptions of what you specifically want then I've seen it produce output that is a reasonably good starting place. If you don't want it using overly flowery, weirdly deferential language and copying specific phrases from the job description then you can address that by including those instructions in the prompt: "Be concise. Use a confident and conversational tone, and avoid using specific phrases from the job description."

The problem is most people aren't going to spend 20-30 minutes creating a really customized prompt. But people who can't write a good prompt probably also weren't going to handcraft a great letter either. Mainly because of the investment of time it requires to create a fully bespoke piece of writing by hand for each place you want to apply.

Another thing I want to mention is that it seems that many people don't realize yet that if you don't like the initial output you can give feedback and ask for revisions. E.g. "Can you make it shorter", "Eliminate flowery adjectives", etc or even just "Can you show me some different variations of this letter?"

If you're writing a lot of these, it's worth putting in 20 minutes or so to make a prompt that creates output that doesn't suck but which you can easily reuse for other companies and roles with minor tweaks.