If you climb improperly, yes, it's like doing a bunch of pull-ups.
But climbing properly, you try to use your legs to push yourself up, your hands and arms to keep you attached to the wall. This allows you to climb a lot more, because your legs are a lot stronger.
As the sibling comment says, you're not supposed to be doing pull-ups. It's more like climbing a ladder. You use your legs for lifting. Your arms are mainly for keeping yourself from falling away.
It still develops strength in the arms and shoulders. But hard problems are more about balance than strength, figuring out how to use limited friction at very specific angles so that you can complete a move.
That said... watch speed-climbing some time. It's a very full body activity, literally leaping from hold to hold. It looks like magic.
