AMP is all fine, but most of the newspapers are lured to implement it by Google with promise of higher page rank on its search results. It all ends up in a walled garden and when the publishers are reliant on it, Google will control the chain of supply. Disgusting behaviour. Just waiting for more people to lobby in EU so they get another fine.
I believe that it routes them into a proprietary cache by google. Taking page views from the provider or at least that is my understanding of it.
I can't speak for everyone, but my objection to the AMP cache is that it's just another way for Google to collect your browsing data.