When you get to be a professional cyclist, the team will have a man who puts new tyres on rims for you.
The statements about reducing bodyweight for hills is very true, but all a professional cyclist needs to do with his arms is hang on to the bars and lift a water bottle to his mouth.
Non pro cyclists will get enough upperbody training from the everyday labours of his dayjob and house cleaning. This is a false statement because as has been talked about earlier this year, many newbies have difficulty getting a new tyre on a rim.
To do the riding around on the road, increased arm strength is not required. It is the mechanical work which requires some arm power.
NB. There is nothing worse than at the end of a 200 Rando, not being able to lift your bike onto the roofrack and asking a sensible guy with some arm muscle to do it for you.
So as a conclusion. No need for weights. Pressups and some chins will suffice.