>Isn't the whole promise of AI tools that they just work?

No, not at all. Like with pretty much any development tool, you need to get proficient with them.

>what skill am I missing out on

At this point, it seems like pretty much all of them related to generative AI. But, the most recent of them that I'll point at is: tooling tooling tooling, and prompting. But the specific answer (to answer your "exactly") is going to depend on you and what problems you are solving. That's why on tries not to fall behind, so you can see how to use tooling in a rapidly evolving landscape, for your exact circumstances.

>I think I'm a reasonably good communicator in both voice and text, so what skill am I failing to train by not using LLMs right now?

You know how to achieve something you will use different words with different people? You don't talk to your spouse the same way you talk to your parents or your children or your friends or your coworkers, right? You understand that if you are familiar with someone you speak to them differently if you want to achieve something, yes?