Nowadays that's hardly an issue because SSE is multiplexed over HTTP/2 and HTTP/3 alongside regular requests. In some ways, SSE is today more efficient than websockets too because if you use it with HTTP/3 then you're avoiding head of line blocking which is still an issue with websockets.

(Also in practice many places websockets are not even supporting HTTP/2 because on the server they were implemented on top of raw sockets which is no longer possible with HTTP/2)