In reply to JVard:
Hi. Weight training can definitely be beneficial. A good program will allow strength and power development in all commonly-used ranges of motion (and as Shark suggests you can make up your own exercises to suit for less commonly-used actions). It also provides opportunities to identify and correct specific areas of weakness / under-activation. For instance there are far too many climbers who are ultra-shredded yet horribly kyphotic and stiff: weight training requires that these issues be addressed, as the immediate alternative is injury. Climbers on the other hand can 'work around' these movement discrepancies while kidding themselves that they're moving well. In weight training, correct form is key, and it offers a kind of therapeutic discipline in regard to healthy movement.
As to whether it's going to make you a much better climber, it kind of depends on what your current general weaknesses are. Is it your fingers, or your core, or your head? Weight training is only going to directly transfer to one of those. If your weaknesses relate to a lack of absolute strength and power, it could bump up your grade significantly. For most people, doing lots of climbing is going to yield the greatest gains, bearing in mind some of the physiological principles of training. But clearly there's a ceiling to the gains to be made from only climbing, where a certain area is letting you down.
Anecdotally I climb very little at the moment, maybe once or twice a month, focusing almost entirely on weight training for now. Despite the bodyweight gain associated with the training and nutrition (up about 8kg from this time last year) I'm still bouldering at the same level or a bit higher. Gives me hope I may actually not be a bumbly, one day
Luke