How can I connect VS Code source control with a GitHub profile to directly push, commit, and sync changes and even push the repository?
Committing and syncing with the git repo is easy if we open the proper repo directory to push.
VS Code should "just work" (see docs) as long as you've configure your git remotes. You can do that via commandline with the git remote commands. If you want to add a remote, use git remote add <name> <URL>. If you want to modify the url of an existing remote, use git remote set-url <name> <newurl>.
For docs, see https://git-scm.com/docs/git-remote and https://git-scm.com/book/en/v2/Git-Basics-Working-with-Remotes.
I don't know what you mean by "publish" (in contrast to "push"). If you mean publishing a GitHub pages site, that should work automatically when you push changes to the branch that your GitHub pages site is configured to track.