I'm really wondering if/why Grammarly still has a business case, in this age of LLMs.

Mainly because (cloud-hosted, bigger & smarter) LLMs probably offer no added privacy benefit. Self-hosted models are dumber, neutered and unpredictable.

It's beyond me how they still refuse to offer an official API.

Gemini subscription costs as much, allows me to do much more, and I can call it from vim or emacs without some arcane acrobatics

They used to have an API, which they removed about two years ago or so. Until then, there were a bunch of 3rd-party Grammarly plugins, including a pretty decent one for VSCode.