Fast websites stayed fast, and slow websites stayed slow. They also gave out tons of free "speed test" tools to make it easy for developers to test their speed, and improve it. They even made apache and nginx plugins that would auto-optimize assets as it served them! And still basically nobody used them.
Plus it's not just about "faster than XXXms". What is that time measured to? Time to text on the screen? Can they game it by lazy-loading images, then videos, then ads, then 6mb of other javascript and social networking stuff and a comment system and...?
You could try a "size limit", but then you penalize asset heavy sites (like high-resolution image sharing sites, or video sites).
So their solution was to make a very restrictive system where you are forced to do things "the right way" (for at least one definition of "right"), and to pump that up in the search results.
Plugins were made for common blogging and news platforms, and now a portion of the web loads significantly faster for many, and I think that's a win.
It's not perfect, but it's at least the first thing that I've seen google try in this area that is actually working.
Obviously there is benefit to Google as a company in that now results that are served through their AMP system are faster than those of some of their competitors, and AMP gives them a nice easy way to pull some structured information from an article for things like the carousel or other non-search offerings, but I genuinely don't believe that was the main motivator, seeing as Google has years of examples of failed attempts to "fix" this problem (though maybe I'm just not cynical enough!).