That was once the case, but no longer. Third-party WatchOS apps can work without a phone present, up to being installed directly from the watch's app store. They can definitely do independent networking, but there are still some restrictions, eg they can't do it when backgrounded, and websockets are pretty locked down (only for audio-streaming as per Apple policy).

I reckon the lack of general-purpose websockets is probably the issue for a system based on Phoenix LiveView.