If the ads are just brought in as a stream of text from the same endpoint that's streaming you the response you're wanting, how can that be blocked in the browser anyway?

Another local LLM extension that reads the output and determines if part of it is too "ad-ey" so it can hide that part?

It will depend on how they implement the sponsored content. If there are regulations that require marking it as sponsored, that makes it easy to block. If not, then sure maybe via LLMs.