How can I disable any sort of on-the-fly diagnostics in eglot?
This does not seem to be possible by customizing eglot-ignored-server-capabilites. Disabling flymake-mode using a major-mode-hook also does not work.
Disabling flymake-mode on eglot-managed-mode-hook should work.
I ended with the same situation, Where I wanted to use flymake directly with eslit wihout eglot. This code worked for me
(add-hook 'eglot-managed-mode-hook (lambda () (remove-hook 'flymake-diagnostic-functions 'eglot-flymake-backend) (flymake-eslint-enable)))