Nginx does have a module [1] and a corresponding configuration option to scan outgoing headers for Link header preload directives, and once it has learned of a preload being declared by a resource, it will push that resource thereafter. Nginx talks about the justification for this feature, where they too admit that statically configuring pushes in the server config is not terribly useful -- it's quite often the wrong place to specify relationships between resources.
[1] https://www.nginx.com/blog/nginx-1-13-9-http2-server-push/
And if you think about it, static pushes in general have very limited usefulness, almost non existent. Imagine when some url becomes popular and almost all of the requests to that url come from people who never visited the website before. It would make sense for a web server to learn what kind of resources clients request with that url and start pushing those resources to people ahead of time.
Similarly, it should be the backend behind the reverse-proxy that knows what's the page that has been just rendered, and knows about the user's session (is it brand new, or maybe it's not new, but still needs to push things because it's too old, and since then that particular page's background changed, etc.).
And in case of an Angular/React/SPA thing, then the "bundler/compiler" should create a list of things to push for various URLs. Or the Angular/React team should talk with the Nginx team to figure out how to speed up things. (In case of SSR - server side rendering - the NodeJS server can emit the necessary Link headers, for example.)
Proxies are already complicated as is. Caching proxies more so. (Think of how Varnish has a - probably Turing complete - DSL to decide what to serve and/or cache and when, and how.)
The HTML parsing approach is probably great from a 80% of the benefit for 20% of the effort on small-scale websites (i.e. majority). A super accurate inference engine might use deep learning to train what to serve on a very personalized level if you have a lot of users & the CPU/latency trade-off makes sense for your business model (i.e. more accuracy for a larger slice of your population). A less accurate one might just collect statistics in a DB & make cheap less accurate guesses from that (or use more "classic ML" like Bayes) if you have a medium amount of users or the CPU usage makes more sense and you're OK with the maintenance burden of a DB. It's a sliding scale IMO of tradeoffs with different approaches making sense depending on your priorities.
That said a simple heuristic like "after requesting an URL the server got these requests on the same HTTP/2 connection in less than 1 second, and those were static assets served with Expires headers" could work.
By "caching layer" he probably meant a proxy or load balancer level cache not the user agent cache. It would make total sense for a load balancer to statistically discover relationships.
That being said, I fail to see how in the general case, setting static headers in the server software's config for Push is useful [2][3], and wish that more implementations converged on a common way of describing what to push [4], so that tools could be built around discovering dependencies, and around interpreting that manifest to execute push.
[1] https://docs.google.com/document/d/1K0NykTXBbbbTlv60t5MyJvXj... [2] https://news.ycombinator.com/item?id=14077955#14081237 [3] https://news.ycombinator.com/item?id=12719563#12722383 [4] https://github.com/GoogleChromeLabs/http2-push-manifest