To be fair, the BBC is institutionally anti-cyclist, so they may have mis-represented the "Dutch Reach".

I can't see why it's not taught and used everywhere as it encourages and facilitates the checking behind you when opening a car door. Rather than focussing on "left" or "right" hand, I find it more useful to just always use the furthest hand from the door so the same idea applies if you're driving or a passenger.

I think it depends on the teacher, but mine didn't teach it either. However, I have been taught from a young age to watch out before I open the door, which is still very relevant even if you're not in the driver's seat.