Ideally, I don't want any server. Just other Bitcoin nodes.
Discussion
Note that on mobile it might be impossible to guarantee that your background process can run indefinitely. The OS will kill it from time to time based on resource constraints.
You might also want to check out compact blocks / Neutrino from Lightning Labs.
Yes, but that's in the past. If you set up services correctly, Android will keep them running. You have to keep them lean and that's my concern on just embeding Bitcoin core an app. Ideally, we would just use Android's native service infrastructure.