This question is but a specific case of the problem of evil, a matter which Christian apologists have responded to many times. I won't offer a lengthy discussion of the problem of evil here, but I'll offer a brief logical & theological refutation of this specific argument.
Logical argument
The argument is at its core attempting a reductio ad absurdum - endeavoring to show that, given xyz evidence, it is absurd to believe that God exists. However, what is presented here is not a valid reductive argument--the flaw is subtle, but crucial.
To construct a reductio ad absurdum argument against God one must begin by accepting, for sake of argument, that God exists. This is the way the first premise is formulated for any reductio ad absurdum--one must start by accepting the truth of the claim in question, in order to show via the absurdity that follows that the original premise should be rejected.
This reductive argument tacitly assumes--as most atheistic arguments do--that there is no afterlife, there is no eternity, and so any injustice not cured in this life will never be cured. This assumption destroys the reductio ad absurdum--to state the premises formally, this argument approaches the problem as follows:
P1: God exists
P2: There is no afterlife
The best the reductio ad absurdum can do is show that these premises cannot both be true; since no Bible-believing Christian would accept premise 2, the reductive argument is entirely ineffective.
The moment we open the door to an afterlife--especially an afterlife that is far greater in scope & duration than mortal life--theological explanations for human suffering in mortality abound. Atheists who do not wish to engage with theological arguments should not attempt reductio ad absurdum arguments against the existence of God, because they'll have to start by assuming (for sake of argument) that God exists. Once that premise is on the table, theological arguments are fair game, and the reductive argument will progress no further without responding to theology.
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Theological argument
God is certainly capable of healing limbs that have suffered permanent damage (examples here and here), but most of the time He chooses to wait before doing so. The maladies of this life are healed in the resurrection (see 1 Cor. 15:54); they are not always healed prior to that time.
The question then becomes why would God allow His children to experience temporary pain? (and mortal life starts looking extremely brief when stacked up next to eternity)
Any parent knows it would not be loving--in the long run--to shield their children from everything painful or difficult. Mortal parents certainly try to protect their children from injuries that will affect them through all of mortality, and our Eternal Parent offers protection from injuries that will affect us through all eternity, but a loving Father who wants us to grow will not withhold from us the very refiner's fire that will allow us to develop. I discuss suffering more extensively in this video on my channel: The Parable of Ana and the Volcano.
To borrow an idea from Michelle Craig, challenges are part of the reason we are here.
Spencer W. Kimball effectively illustrated why a loving God should not remove every obstacle:
If all the sick for whom we pray were healed, if all the righteous
were protected and the wicked destroyed, the whole program of the
Father would be annulled and the basic principle of the gospel, free
agency, would be ended. No man would have to live by faith.
If joy and peace and rewards were instantaneously given the doer of
good, there could be no evil—all would do good but not because of the
rightness of doing good. There would be no test of strength, no
development of character, no growth of powers, no free agency, only
satanic controls.
Should all prayers be immediately answered according to our selfish
desires and our limited understanding, then there would be little or
no suffering, sorrow, disappointment, or even death, and if these were
not, there would also be no joy, success, resurrection, nor eternal
life
...
Being human, we would expel from our lives physical pain and mental
anguish and assure ourselves of continual ease and comfort, but if we
were to close the doors upon sorrow and distress, we might be
excluding our greatest friends and benefactors. Suffering can make
saints of people as they learn patience, long-suffering, and
self-mastery.
...
I would likely have protected Paul against his woes if my power were
boundless. I would surely have healed his “thorn in the flesh.” And in
doing so I might have foiled the Lord’s program...Paul many times could have lost himself
if he had been eloquent, well, handsome, and free from the things that
made him humble.
...
With such uncontrolled power, I surely would have felt to protect
Christ from the agony in Gethsemane, the insults, the thorny crown,
the indignities in the court, the physical injuries. I would have
administered to his wounds and healed them, giving him cooling water
instead of vinegar. I might have saved him from suffering and death,
and lost to the world his atoning sacrifice. (source)
My own experience gives me the conviction that God does hear & answer prayers, but He does not always give the answer I want. God could give us what we want today, but He loves us enough to offer something eternally better.
Addendum--response to questions
Why would God heal other ailments but specifically exclude healing (in this life) for those who had a part of their body removed?
I provided an example (linked) above that God has performed such healing and has not unilaterally excluded such injuries. That said, there are so many ways a human being can be injured it would be unrealistic (and contrary to the Biblical principal of faith) to believe in God if and only if He demonstrates His ability to heal at least one instance of every possible form of injury. I'm unaware of God ever raising from death a person who died while skydiving, but that does not lead me to conclude that He doesn't love these people or lacks the ability to heal this precise form of injury.
If God can raise the daughter of Jairus, the son of the widow of Nain, Lazarus, Eutychus, and Dorcas from death, I see no reason why He couldn't raise a skydiving victim from death if He saw fit to do so.
Does the reductive argument really require premise 2 above?
Yes, consider what the argument would have to say otherwise. If we grant the possibility of an afterlife--for sake of math let's say the afterlife has a duration of 10^500,000 years (of course if we accept an eternal afterlife it's much longer than that)--the question becomes, how is it fair that:
- Person X suffers a grievous ailment for 50 years out of 10^500,000 while
- Person Y suffers a grievous ailment 0.5 years out of 10^500,000
The force of the original argument is lost.
Why doesn't God just make people already in a state of eternal life?
A more complete answer to this question would require a discussion of what is meant by eternal life--and I personally have a very high view of the term--but the short answer is:
- God is a loving parent who wants us to develop--life is about becoming, not filling a checklist, and the process is important (would any parent want their children to be born already as adults?)
- The objection that this means God is not Omnipotent is bogus--it's the same form of objection as claiming God is not Omnipotent because He cannot create a square circle. A square circle is a contradiction in terms, it's a meaningless combination of words. So too is the idea of a spiritually immature recipient of eternal life.
