Both downsides are fixed by using http/2, right? Heartbeat seems trivial to implement using HTTP/2 PING frame. And you already mentioned the stream sharing, which IMO is much better than the state of each tab on the same website opening its own TCP socket, pretty much a resource exhaustion hell (browsers didn't implement websockets on top of h2 last time I checked).
Websockets do have an advantage over SSE: they get binary streams instead of text.