From what I understand websockets are great until you have to load balance them. And then you learn why they aren’t so great.
You need to scale up before your servers become overloaded, and basically new connections go north to the newly brought up server. It is a different mentality than scaling stateless services but it isn't super duper hard.
Ideally it is based on the rate of incoming connections, but so long as you leave enough headroom when doing the stupid simple scaling rule you should be fine. Just ensure new instances don't take too long to start up.
I guess with WebSockets, if you choose to send messages from the client to the server, then you have some additional work that you wouldn't have with SSE.