Although AMP is just a subset of HTML, if your site passes AMP validation, it gets cached by Google's CDN. I believe that's what makes AMP pages feel fast.
You can certainly write HTML that doesn't do anything of the things that AMP prohibits, but if it's hosted on a slow server, your users won't get the lightning-fast experience. By caching the pages, Google takes the quality of a publisher's host out-of the equation. (In fact, since the pages are being cached anyway, publishers probably have less of an incentive to invest in speedy server rendering.)
AMP is not a subset of HTML; AMP replaces many HTML tags with its own proprietary custom elements, like amp-img, while banning normal, standards compliant img tags. It also has a bunch of vendor-specific tags, like amp-instagram and amp-twitter (but no amp-flickr or amp-vk), so Google gets to be a bit of a kingmaker in what content gets bespoke tags. Ads have to go through amp-ad, where Google also controls which advertisers are allowed on AMP pages, so you're allowed to show ads from the Doubleclick network, but not from The Deck or Carbon networks. Oh, and Google rehosts your AMP pages, so you don't even get to see traffic data to your own content.
This is a massive landgrab by Google that only incidentally solves a real performance problem on the web.
You can certainly write HTML that doesn't do anything of the things that AMP prohibits, but if it's hosted on a slow server, your users won't get the lightning-fast experience. By caching the pages, Google takes the quality of a publisher's host out-of the equation. (In fact, since the pages are being cached anyway, publishers probably have less of an incentive to invest in speedy server rendering.)
(This is my understanding. I don't work on AMP.)