Perhaps I'm wrong, but I believe HTTP streaming is for chunking large blobs. I worry that if you use this pattern and treat streaming like a pub/sub mechanism, you'll regret it. HTTP intermediaries don't expect this traffic pattern (e.g., NGINX, CloudFlare, etc.). And I suspect every time your WiFi connection drops while the stream is open, the fetch API will raise an error as if the request failed.
However, I agree you probably don't need WebSockets for many of the ways they're used—server-sent events are a simpler solution for many situations where people reach for WebSockets... It's a shame SSEs never received the same fanfare.