An adblocker is a good counter-example: it could work with an API that tracked the page changes and stopped e.g. JS or images inserted by the extension from loading. Even when editing the DOM, it has no need to load anything externally. Sounds possible, but I'm not sure how big the overhead of extending the DOM like that would be.

You probably could still create interactions with on-site JS that leak data sometimes, so it wouldn't be perfect, but that's page-specific and a lot more work.