This definitely sounds like a bad battery to me. If it's stuck on 1%, that's textbook for battery data failure.
The phone needs information about how much charge the battery has left and if there is a failure of the battery management system (which is part of the battery) that data is often unavailable. So the phone may have no idea how much charge is left, hence why it's stuck on 1% and if it doesn't think there's enough energy to complete boot, it won't.
There are other things this could be, but battery makes the most sense.