The EventSource API (the browser "client API" for Server-Sent Events) leaves a lot to be desired. While I am a maintainer of the most used EventSource polyfill[1], I've recently started a new project that aims to be a modern take on what an EventSource client could be: https://github.com/rexxars/eventsource-client.
Beyond handling the custom headers aspect, it also supports any request method (POST, PATCH..), allows you to include a request body, allows subscribing to any named event (the EventSource `onmessage` vs `on('named event')` is very confusing), as well as setting an initial last event ID (which can be helpful when restoring state after a reload or similar). And you can use it as an async iterator.
I love the simplicity of Server-Sent Events, but the `EventSource` API seem to me like a rushed implementation that just kinda stuck around.