I like training static and slow movements for climbing. I feel it translates the best to actual climbing.
I usually find a pull up bar and just do a lot of different grip width variations. Some narrow grip, some shoulder width, and some as wide as I can. All of them are done slowly and controlled, so I don't jerk myself up. I start from a complete dead hang position, slowly pull myself up, and even more slowly lower myself back down.
I also try to pull myself further up, so that I am in between a regular pull up and muscle up.
Training one arms is probably my favorite though, it also really requires very minimal gear, basically just a pull up bar.
Anything body weight and slow movement helps me, so pull ups, push ups, handstands, etc.
Flexibility in my opinion is extremely underrated. I think being flexible can be one of the greatest tools in climbing.
I myself am not flexible at all, but my friends are, and the difference between how we climb is so noticeable. Moves that are so difficult for me are just easy to them.
You dont have to be flexible to climb, but, it would make a huge difference if you were.
It can equal stronger climbing, but not necessarily better climbing.
The problem with being very strong is that you start to neglect technique, because if you can just power yourself to the next hold or through the move, why would you do it in a more technical way? Even if you try to be more technical, it can be very hard to get rid of the habit of powering through moves. This is something many climbers struggle with, including myself.
On the other hand, if you are a technical climber, but are weak in terms of physical strength, training your forearms can help quite a lot by balancing you out a little bit. So you would still have the technique that you have developed until now, and you would also just be physically stronger.
So I would say that it just depends on your current situation. Are you more technical, or are you more powerful?
Rock climbing doesn't really get you big and bulky. It can make you stronger though, just without the massive body size.
There are some people who tend to grow and bulk up faster than others, at least to a certain size. They may get some more muscle mass than others, but for the most part, you muscle mass won't really increase that much.