When looking around online to see how other people handled climbing ledges, you tend to get one of two extremes:
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Simple system that uses objects flagged specifically as an object meant to be climbed on. (Or hidden colliders on edges that do the same.) This seems annoying to do because you have to have your climbing objects separate from all the other world implements.
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A completely procedural climbing system that does a bunch of geometry reconnaissance and acts accordingly. Games like uncharted 4 or assassin's creed do this. It seems to do well in replicating genuine rock climbing with hand holds and foot holds and whatnot. It seems a little overkill in some scenarios.
Personally I would like to do something in between these two extremes. Something that is procedural enough to allow you to catch a simple ledge, hop up, let go, shimmy left and right, and call it a day. A game I'm like 90% sure does this is Overgrowth. You can catch almost every ledge you see, but you don't have all these other obscure use cases. It keeps it relatively simple, but it's just procedural enough to allow climbing where ever you are.