Server push sends responses to requests, only these requests are “promised” by the server. So, for every event pushed, the server would have to send at least three frames: promise (the synthesized request), response headers, and response data.

Also, you would still need something else for HTTP/1.x. Even an HTTP/2 client does not have to support server push, and can even ask the server to disable it.

Why do you think server-sent events are not HTTP-y? I found them as elegant as modern Web technology gets.