"I want that site not a google cache of it" might be true for you, but for me, I don't want the site so much as the article or product listing, and for most users, they don't know that it's a google cache of it.
I think AMP provides the tools for a good UX but many sites don't provide it because they want the user to go to their site, because they mistakenly don't think they can get enough ad revenue or CTAs to get the user to sign up for mailing lists or add a product to a shopping cart on the AMP page. These are supported. So they only show an excerpt of the article on the AMP page and you have to go to the site in order to get it.
I think AMP provides the tools for a good UX but many sites don't provide it because they want the user to go to their site, because they mistakenly don't think they can get enough ad revenue or CTAs to get the user to sign up for mailing lists or add a product to a shopping cart on the AMP page. These are supported. So they only show an excerpt of the article on the AMP page and you have to go to the site in order to get it.