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
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