I'm so glad others have noted this issue with unweighed baggage. A couple of years ago on a BA flight from EWR back to the UK, I was appalled how much hand luggage (by hand luggage I actually mean big heavy suitcases) was being allowed on - especially as you can check at least 1 23kg suitcase into the hold.
BA didn't seem to be enforcing their own rules, and when we boarded the 777, crew were really struggling to find spaces for the passengers stuff. I even saw 1 passenger dragging 3 items on board.
I'd be more than happy to see this policy strictly enforced. I've seen easyjet crew enforce this effectively on passengers as they board, so I'm sure BA could.