HTTP/2 doesn't fix the former downside. It doesn't give you a general bidirectional stream, so you can't implement flow control. I'm not aware of a browser API for sending (or inhibiting) HTTP/2 pings - is there one? In the absence of one, you also can't use HTTP/2 to implement an application-level heartbeat.
To expand on this a bit, a problem i have run into with SSE is that the socket is open, some part of the client (the browser or HTTP client) is reading data, but it is not being processed, because the actual application code on the client is stuck or broken or something. To deal with that, you have to be able to drive feedback from the application code on the client to the server.
