This reminds me of https://slrpnk.net/post/280323, about doing similar things with old Chromebooks. I definitely think there should be a movement toward using random older devices for hobbyist projects. I mean, there's nothing actually special about the RPi, just that it's small and enough people use it that they can pre-prepare software for that specific hardware. But it should only be a little more work to make just about any computer do the things that people use the pi for. I think the possibility of a bigger movement is there!
Regarding the battery overheating, correct me if I'm wrong but I thought generally the electronics would keep that from happening. Of course, keeping it always on charge would probably degrade the battery life of the device, but once a phone is on server duty that's not so much of a concern anyway.