Thanks for the responses, everyone!
I cannot imagine a couple of people in their seventies who aren't in very good shape managing to climb or descend that White Mountain with a mountain bike. They might be able to do the first 5 miles where the road isn't as bad, but that final climb....
Good point.
I guess we have two choices when we get off the bikes at mile 5. Option 1 would be to lock the wheels of the bikes to each other, leave them a few feet off the trail, and unlock them on the way back down. Option 2 would be to attach our daypacks to the bikes and walk/push them to the summit. I'll probably go with option 2 if they're still feeling good at that point.