No, it does not happen, nor will it ever because that's fundamentally not how AMP caching works. A site doesn't pick an amp cache like they pick a hosting provider, with some sites cached on cloudflare-amp while others are hosted on google-amp. All amp content is on all amp caches. Google search is one AMP client, and it uses google's AMP cache. If, for example, bing wanted to serve AMP pages, they would need to choose an AMP cache to serve them from, and cloudflare's offering would be an option. One of the problems that AMP is trying to solve is individual websites hosting on insufficient architecture. If each site was allowed to choose what AMP cache it was served from, that would defeat the purpose. It's up to the client to choose the cache it uses, to ensure a consistent speed.

That's pretty much the point I was making. The wording from Google to publishers doesn't make this clear.

It is the implementation they chose, and it does have speed benefits. But, an edge cache that allows the end user more control, their own domain, an no hijacking header is possible, and would provide a notable speed benefit. If there were a choice between the two, publishers would likely pick the latter.

Publishers had a choice, and they overwhelmingly chose to host their sites on infrastructure that was too slow. That's why the AMP project exists, to take choices like that away from publishers, because they consistently make the wrong choices when you give them one.