In reply to cas54321:
Yes, you're being hugely over cautious.
Lots of people will have been taught to lock off a belay device by putting the dead rope down one side of the body, where it is impossible to put two hands on. Perfectly safe.
Locking off centrally with two hands on the dead rope is relatively recent.
Even then people are still taught to use their second hand to pull the live rope to make it easier for the dead rope to be pulled through the device.
As others have said it is impossible to lead belay with two hands on the dead rope even though a lead fall is harder to hold and may involve a moving rope rather than a static, locked off, top rope fall that involves no movement of the rope at all.