If God, aka "the infinite," healed amputees, we'd know God existed, then we wouldn't have to believe in God. If we can point at proof of God, we don't get the enlightenment that comes from believing in God.
Then the infinite would have no purpose for existing, so the infinite would disappear in a puff of logic. There is a theory which states that this has already happened.
Prayer doesn't work because the infinite is immune to our desires. the infinite may care about us, but if the infinite wanted us to have $1000 we'd have it already. The fact we don't have it means the infinite doesn't want us to have it or doesn't care.
Scientific studies in fact have proven that prayer helps, but it only helps the people who believe that prayer enables them to change reality. Believing one can change reality is its own reward. Belief has psychological, emotional rewards, even belief in a jug of milk.
Psychologically, belief has a lot to do with choice. I would love to choose to believe in a jug of milk. The powerful have power of choice over what they believe. Pity those who believe they have no power of choice over what they believe, for they are the weak.