I don’t leave syncthing running on my android devices because it is a battery drain. I propose that the desktop versions of syncthing allow remote devices to be designated as “mobile devices.” When changes are seen, a push notification is sent to android that starts the syncthing service, syncs the changed files, and then stops the syncthing service.
The Syncthing protocol is essentially “push notification” style already. I suspect the battery drain comes from Syncthing not properly sleeping for long enough to let the mobile CPU shut down etc, as there are a lot of background things ongoing.
“Real” Android-level push notifications, handled by Android/Google when the app is not even running, is not something that fits into how Syncthing works unfortunately.