The issue with it staying running could be the contacts tack-welding themselves together, not sure what to do about that. It would explain why it doesn't happen when you push harder, because you're either getting enough contact to avoid hot spots, or breaking the tack welds that formed, like when you have to hit it a few times to get it to stop.
The issue with the rubber on the wheel looks like it was slightly misaligned with the truggy at some point, and this created a sideways force on it that pulled the "tire" aside. Maybe you can get it back into its groove, maybe just a new wheel.