In short, at a certain level, climbing should typically be supplemented with some other form of work-out. For me, the V5/6 range is where I needed to add in supplemental training. Rope climbing can provide decent cardio, and bouldering is good for pulling strength. But there are gaps that need to be filled, which vary from climber to climber.

I generally boulder 2-3 days per week (and add one day of volume rope climbing), deadlift one day a week, around 5 sets of 5 reps (after warm-up), and try to do push-ups, shoulder/scapular work 2 days per week. I've worked up to one arm pushups, and find these much better for increasing the range of motion in my pecs/shoulders. Shoulder training consists of handstand push-ups, planche progression, and scapular pull-ups. This supplemental routine was key for me to correct shoulder imbalances and heal tendonitis.

I also have been following this warm-up religiously for a month or so and am gaining flexibility for the first time in my life (not needed for everyone, but I desperately needed to increase flexibility for general, all-around athleticism): improvement.https://www.youtube.com/watch?v=X2H-RZ8kNRg. If I'm focused, takes me around 20 minutes or so to execute, and subsequent warming up on lower graded problems takes much less time.

Weighted dips, muscle-ups, pistol squat progressions, and front/back levers are thrown in randomly/periodically. I also do a couple sets of low rep, weighted pull-ups maybe once or twice a month to see how I'm progressing (and I do see gains every time I attempt it). I hangboard and campus sporadically, to assess and try to improve muscle recruitment.

Not really advocating my specific training, as everyone is different. I personally avoid structure and systematic recording of my workouts like the plague. Working everything to failure and experimenting with alternating my workouts from pre/post climbing to separate non-climbing days is part of my process. Just wanted to throw in my anecdotal evidence, as this training took me from projecting V6/7 (indoors) in many sessions, to sending V6/7 in one session, with the occasional V8/9 project send. My gym seems pretty comparable to outdoor stuff, as I completed my first V7 outdoors in one session at the same time I was projecting V7 indoors. Really need to get back out there to reassess soon. This improvement took about 6-8 months.

Climbing by itself was beneficial to me for a full 2 years before I began to notice imbalances, and then started throwing in supplemental training on an as needed basis. The above routine is personalized for my specific needs. I have trouble dropping weight, so getting stronger to carry my weight worked for me. Other people might want to run, bike, etc. to get leaner (and perhaps getting leaner instead of stronger will help avoid imbalances too, idk). I am also leading outdoors around the 12+ range, so endurance/efficiency is something I need to work on to bridge the gap between my bouldering and leading ability. If that's where you want to improve, you likely need to focus on a different skill set, but at the moment, I'm satisfied with getting strong/powerful for specific moves and slowly filling in the gaps for improving lead climbing.