Bike for Life is a great inspirational book but it's not that good on exercise physiology. Weight training won't help your climbing much. The muscular systems used to do 10 reps of weights are different from what you use to do hundreds of pedal revolutions on a climb.

I do weights in the winter and try to maintain my gains through the season. But inevitably I lose the ability to push the weights through the season, especially as I stop lifting around peaks for A races. I'm down to less than half of what I was doing in February. But I'm climbing significantly faster, and I'm only a few lbs lighter so it's not due to lower weight.

Where weight training is good for cycling is in the supporting muscles- core and upper body. And it's useful for non cycling fitness, bone density, etc. but that isn't making one faster on climbs.