The puzzling part is fun, because you are constantly learning new ways to use your body. See how to balance, how to move around, etc. I have found that dancing gives a similar learning challenge. Especially the more free-form dances like salsa and bachata. It’s fun learning new moves every week during the lesson, and then try to see if you can put them to practise during a party.
And don’t worry about beeing to stiff. If you can balance around boulders, you can get your body to move around for dancing too. Just takes some practise. I currently do both, and feel like I lack dexterity more for the climbing than for the dancing.
And unlike most of the hand-friendly options mentioned already, you do have to use your hands and arms a lot. Just not in a way that puts any stress on them.