> unless I suppose there's an lip-mode for TeX

https://emacs-lsp.github.io/lsp-mode/page/lsp-latex/

> for autocorrection when sending mail or writing random documents

Complete-at-point is integrated with company-mode, of which I use lsp as a backend to serve suggestions. So ... In a way, yes, I do get spelling suggestions through the same mechanism that lsp works.