I test enterprise software and I can tell you there are always a list of "known" bugs in the backlog that never gets fixed unless an existing customer asks for it. This is the case in every large org I've worked in.
You should be able to use the Smart HTTP protocol with HTTP Basic Auth (https://git-scm.com/docs/http-protocol) to push a locally-created commit. Forgejo supports CORS here, but I don't know whether GitHub does.
That's not to say one couldn't still do what you're describing via other headers, I'm just saying "<input name=username><input name=password>" won't get it done