The hammer example made me remember something. I did some Aikido long ago, and the instructor spent quite a lot of time showing us how to grip things like sticks. As I remember it, instead of the instinctive way of just forming a fist around it, we should instead start from the little finger, wrapping the fingers one by one, but letting the index finger actually rest more along the handle than wrapping it. That way, supposedly, the grip is just as good, but more flexible and the index finger can help with control.
Most martial arts teach something like this.
Typical civilian relies too heavily on the index finger when grabbing for instance your arm, and that makes it easier to twist out of the grapple, by using the shoulder and the bicep to lever out along the line halfway between the thumb and the index finger. Usually these are stronger muscles than the forearm, possible exception of rock climbers.
index finger is useless for holding tennis racket too.
This is a classic way to teach use of a sword. It's also easy to feel what happens. Compare the feeling when gripping with first two fingers vs 3rd and 4th. With first 2, you will feel tension along top of forearm, whereas with the other 2 it is the underneath of forearm. This affects flexibility, softness, and thus ability to manipulate the sword.