What I don't understand is why they cache the page themselves, and make that the requested version.
It would seem like any cached page from a CDN would be just as quick.
If Google preloaded the page from a non-google server, there would be a few problems. The most important one is that it would violate the user's privacy. That server would have in it's logs a request from a user that didn't click on the article, and could intuit information about the user's searching behavior from that request.
Less importantly, without the page being served from Google's cache, there is no way to guarantee it's actually a valid AMP page. The only thing Google knows is that it was valid AMP the last time it was crawled, which could be days ago.
And a waste of bandwidth if the user never clicks.
That's one problem.
> without the page being served from Google's cache, there is no way to guarantee it's actually a valid AMP page
Yes there is. You can validate pages as AMP on any CDN. You just can't have any benefit unless they're hosted on Google CDN.