Well that's what I implemented. There are markdown docs for every HTML file and the proxy decides to serve either markdown or HTML based on the Accept header.
I think GP meant on the client, i.e. agent side. As in, you could deploy this kind of proxy in a forward/non-reverse way inside the agent system, so the LLM always gets markdown, regardless of what the site supports.
There is no real reason to pass HTML with tags and all to the LLM - you can just strip the tags beforehand.