I’m not sure if I am understanding your post correctly, but AFAIK you can just hit “.” from your browser on GitHub and it opens an editor (based on VSCode) for that repo.
Not sure if Git is fully integrated in there (although, it being based on VSCode, it should).
Yes, just hitting dot in Github opens up Github.dev which is very similar to this VSCode.dev. So similar that this VSCode.dev lets you get the same experience if you rewrite a Github.com repo URLs and prefix it with VSCode.dev. Today VSCode.dev also supports the same thing with Azure Repos if you prefix dev.azure.com repo URLs.
The "remote repository" experience on VSCode.dev is a little more involved than just "fully integrated git" (it relies on the GitHub/Azure Repos APIs and existing VS Code extensions with Remoting support for some of it beyond just what raw git can provied), but they say in this article that it is extendable and they can and hope to add other providers in the future.
Not sure if Git is fully integrated in there (although, it being based on VSCode, it should).