The mobile app uses the location services in the device to determine if the device is in or out of the circle on the map. That is it, no gps data is transmitted or leaves the phone. Only an event on enter or exit is sent directly to the hub. This requires an active cloud connection (hub and device).
So if you are on wifi, it will communicate your presence on wifi. If you leave the geofence but stay on wifi or transition to cellular, it will send the exit event to the hub.
Accessing a wifi access point name is not a reliable means to determine location. For some, it might be good enough but for others it isn't going to be reliable. On top of it, turning off wifi or rebooting your WAP isn't leaving your house, so how do you tell the difference.
For those that want GPS based presence / geofencing the mobile app is a solid and reliable way to determine if you have crossed an arbitrary line on a map. Other solutions you outlined are simply if you have joined a mesh or network or not. Both serve different purposes.
The mobile app was developed for those that wanted all 3 services, remote dashboards, push notifications and geofencing. Nothing requires you to use the app.
Push notifications are working fine and implemented, but something isn't working on your end, you can try logging out and back in and see if that fixes it.
