That man is descended from apes is an established fact, and that he also has a fondness for climbing seems to be an instinctive behaviour linked precisely to this descent. In the more than five million years of evolution that separate us from our chimpanzee ancestors we have not entirely lost this instinct; it is true, though, that for us it's not so easy to stay hanging from our arms the way apes normally do.
Anyone getting ready to take their first steps in climbing knows it: the instinct is to rely on the arms. You often end up clinging to the rock, trying to stay hooked onto it. Nothing could be more wrong.
Evolution has led us to favour the legs and feet. Standing up requires continuous adjustments of the centre of gravity, which shifts constantly within the area between our feet. These adjustments, made imperceptibly by our muscles, are essential for balance.
In sport climbing (at least for those starting out in this wonderful sport) you don't need great muscular strength in your arms. When the feet have found their right foothold, the arms serve first of all to give us the right balance. They keep us close to the wall, but no weight should bear down on them.
There's no point pulling hard on your arms: after a first effort your muscles will feel like they're about to explode. If instinct has led you to look for a hold very high up, the result is that the blood can't flow to the ends of your arms. In the long run it becomes impossible to hold the whole weight of your body on your arms (unless you do continuous and intense training).
When you are on the wall the instinct is still to grab every tiny crack, paying more attention to the arms than to the legs. It's a primal instinct but it is essentially wrong. You have to calm your fear of falling and focus your attention on the position of your feet and legs.
We can conclude that arm strength in sport climbing (at least at the lower grades of difficulty) is not fundamental.
Notes:
In the first photo the correct position has both feet at the same height and the hands controlling balance and distance from the wall, so that the centre of gravity falls between the two feet.
In the second photo the convex wall pushes the climber outwards (the centre of gravity is between the two feet but towards the outside); instinctively the climber looks for a hold very high up, in an extremely tiring position that he won't be able to hold for long.