Robert Smith gives us more reason to reject the objection that God supposedly doesn't heal amputees. He's discussing healings at Lourdes, but the principles he lays out are more widely applicable as well:
The objections fail, however, in that they exaggerate the gap between cures of grave organic disease and regrowth of a limb. If natural causes can be alleged in the instantaneous cure of organic disease, they can also be alleged in regrowth of a limb, should that phenomenon occur. If a materialistic observer were called to verify that a man's amputated arm was restored suddenly at a spot famous for faith cures, he would not cry out: "Here is certainly a miracle!" He would say: "This observation which up to now is unique leads us to conclude that under certain circumstances which as yet are indeterminate, the tissues of a human limb have the power to reconstitute themselves, in the same manner as the claws of lobsters or of crayfish, and the tails of lizards, but much more rapidly. It is a fact of nature in apparent contradiction with other facts of nature. This apparent contradiction results from our ignorance of the nature of the healing powers of the human body."
The fact, then, that Lourdes stops short of restoring amputated limbs does not point to the operation of limited purely natural forces at Lourdes. Natural forces can be conceived of, at least theoretically, as going beyond these limits and restoring amputated limbs. The cures at Lourdes are not at the theoretical limit of natural forces and therefore fittingly to be attributed only to the action of the latter.
Thus, if the reason why God should stop short of such cures is not clear to us, neither is the reason why natural cures would stop there altogether clear. The fact that no amputated limbs are restored at Lourdes does not imply that only natural forces are operating in these cures. (Comparative Miracles [St. Louis, Missouri: B. Herder Book Co., 1965], 95)
For a response to the notion that alleged miracles are just manifestations of human psi or some sort of unknown natural ability of humans, see here.