I think this is different than server push like WebSockets; this is all about pushing down resources the server suspects you'll need.
Server push, websockets, h2 steams and webtransport are all different things.
Some caching proxies struggle with it though, because they try to buffer an entire response before forwarding it. And it’s 1 way. For something like collaborative editing, crafting a POST request for every keystroke feels wasteful, though it works great in practice.
I use the heck out of SSE to add update notifications to various of my apps because it's just so easy to do. Have it glued to a Redis stream (or pubsub channel) with https://github.com/bonkabonka/sseredis (though that shim looks like it needs a bit of love and a better README.)