I also had some experience in that space, and my conclusion was that what was really hard is handling keep-alive correctly. Some mobile networks will very aggressively flush their NAT entries while other will be fine with a packet every hour or so. And sometimes the NAT timeout changes depending on the server's IP range. Building a solution that is both battery-optimal and that will keep the connection alive is pretty hard.
QUIC (or HTTP/3 with enough low-level control) sounds like a good fit: there is no NAT connection that has to stay alive, but the server can remember the handshake for a quick restart. Doesn't matter if you switch between wifi and LTE. When ever the app is told radio is now available, send a single UDP packet to request new notifications, server sends one of more UDP packets back in response.
