In reply to Kemics:
I've always used pull-ups as part of my training, since before there were climbing walls to visit, and living most of the time too far from the real rock for an evening visit after work.
I've virtually never done a single set to exhaustion. Trying that now (at 63) without a very solid warm-up would guarantee injuries I'm sure.
What I do is a pyramid. 1 pull up, 30 seconds rest, 2 pull ups, 60 seconds rest 3 pull-ups 90 seconds rest etc up to 6, 7, 8, or 9 depending how fit I am, then back down to 1 again. (And if I've gone up to my limit, every set on the way down is a struggle - no pain, no gain they say!). All done palms away. Used to be on a scaffold bar across my loft access, now on the large finger holds on a BeastMaker 1000. Oh, and all those resting seconds are put to good use as well, tidying my room mostly!
I don't do much sport climbing, and none above 6b, but I've never seen the pull-ups as training for that. What they have given me is the confidence to hang around a bit longer, on big scaffold-bar-sized holds, when placing gear on steep trad. And I do have a rough grade equivalence in my head. Pyramid up to 6 = OK for VS, 7 = HVS, 8 = E1, 9 = E2 - never got beyond that.
So there's a slightly idiosyncratic answer to your question. I don't know anyone else who does this but it's worked for me.