Yeah, we're also pushing for SSE for this kind of things.
Starting with out of the box swagger support: https://github.com/OAI/OpenAPI-Specification/issues/396
Shame they are not interested in adding it.
Sadly that makes it a bit of a pain in the ass to manage for sites served as (/ compatible with) http 1.1.
To your point, https://caniuse.com/eventsource
Also there are polyfills for very old browsers like IE.