That's a difficult one to call. 

If all they are responsible for is opening and closing the doors, then you have an issue with what's known as 'Cognitive underload'. Basically it's where an individual's mental capacity is not fully utilised during a task, leading to potential issues like boredom, distraction, and decreased performance.
This individual is probably more alert than someone who open and closes the doors and ends up having a micro sleep between those duties.  However, choosing to watch a video is far from the cleverest way of doing it, knitting though, I'm 50/50.  You can do that activity and still have eyes on the road ahead. 

Not really, a manager can't watch over every single tube driver constantly. Drivers are all adults and are fully aware of their responsibilities.

You either automate the lot or give driving back to the operator and get rid of ATO^▸ (Automatic Train Operation).

Perhaps a suitably trained Alsatian dog could operate the train ?

They would need to bear in mind that dogs are highly organised pack animals like humans, and this would just lead to demands such as the right to demand treats after they have wee-ed in the places provided for them, and the right to work wearing nothing but a leather collar.