SSE is just a long-running connection, so it fits with HTTP/2 pretty well (and HTTP/1 for that matter).
H2 push needs to be linked to an initial response, and I don't think it's acceptable to push resources once the initial response is complete. Also, H2 push is pretty much a request/response pair, so it can be cached, so it doesn't really fit the SSE model.
You're right about websockets though. At some point they'll be replaced with something more H2-like, but it won't be push-based.
Right; Server Push isn't "send the client arbitrary data"; Server Push is "move link/subresource prefetch logic to the server, where you can take advantage of optimizations like single JOIN-queries that give you all you need to output multiple resources, actually outputting multiple responses."
H2 push needs to be linked to an initial response, and I don't think it's acceptable to push resources once the initial response is complete. Also, H2 push is pretty much a request/response pair, so it can be cached, so it doesn't really fit the SSE model.
You're right about websockets though. At some point they'll be replaced with something more H2-like, but it won't be push-based.