They just released an update (for iOS, anyway) that appears to add/fix AirPlay support (I saw the icon but didn't test this), and adds support for Picture in Picture. The PiP support is a little weird in that it doesn't just happen when you switch apps, you have to tap a dedicated non-system PiP button that does it. But it's there! Unknown if it fixes the issue with the stream stopping every few minutes for live events.

Kind of surprised these feature fixes/updates aren't called out anywhere, even the release notes (maybe the app reminded you; I checked the App Store first before opening the app). Anyway: if you don't have your phone set to auto update, or if it takes a while to get around to updates like mine, then be sure to update the app.