Many distributions package tree-sitter-cli, e.g., homebrew. Check yours. You can also download the release artifacts from Github, if your platform is one of the standard ones.

Dependency for https://github.com/nvim-treesitter/nvim-treesitter?tab=readme-ov-file#requirements (I didn't notce this was another repo).

Unfortunately the version in apt is quite old: 0.22.6-6

Yep, that's too old.

I actually had npm and tried npm install tree-sitter-cli, but :checkhealth still gives ERROR tree-sitter-cli not found. How to continue?

Not here. If you need help, ask on the nvim-treesitter support forums. (Of course the tree-sitter-cli must be on your $PATH so that it's available in whatever environment you use.)