anyone else with problems that an ATV3 won't be able to act as a gateway anymore after upgrading iOS/macOS to 13.x/Catalina? Maybe something related like older iOS versions and handoff (like Safari tabs,...)?
Absolutely. I had that exact issue before tvOS 13 came out for the ATV4, and I'm still having trouble since the HomePod OS isn't out yet. If my ATV4 wasn't working before the update, I bet the ATV3 has the same issue if not worse. This was a ridiculously clunky launch given how smooth last year's iOS12 release was.
