That's probably because you have not built a massively scalable system from scratch.
Websockets use one TCP socket for starters with SE you use separate sockets from ingress and egress which is much simpler to scale from an infrastructural point of view.
I clearly said many-to-many, everything else is trivial to scale because only many-to-many requires shared memory atomic parallelization.
You could write a better server in Java than javascript/c(++) but I haven't found any and to be honest the extra complication of the binary protocol doesn't even make me curious enough to waste 5 seconds to google it.
SSE over HTTP/1.1 is the final solution. Everything else is a waste of time for eternity.
Not necessarily, it depends on the needs, but in any case I would not expect SSE and WebSockets to be different in this regard.
I'm not sure what's specifically hard to parallelize with WebSockets. Shared memory may be hard to handle but that does not seem specific to WebSockets which are just regular two way communication channels that happen to start with an HTTP request from the client.