I didn't know about it, AMP site lists only Google, Bing. But I know for a fact that cloudflare has no issues caching AMP sites like any other sites though.

It looks like they killed it last year. The codename was Ampersand.

Creation: https://www.cloudflare.com/press-releases/2017/cloudflare-an...

Death: https://blog.cloudflare.com/announcing-amp-real-url/

I agree with you that users should be able to choose their Amp Cache.