joining this thread late (my first post) ...
If we think that the material world is all there is, and that there is nothing after death, then yes, it would appear that God not healing an amputee (or allowing it to happen in the first place) is terrible and makes no sense. This naturally leads to hating God and/or not believing etc. Completely understandable. And if people are convinced that there is only one life, and nothing happens after this life (or possibly, we have one life and then get judged on where we spend eternity), then i see virtually no argument that could answer the original questions adequately. It would lead to atheism being the only logical conclusion.
However, consider for a moment that this life is just one of many. That is, reincarnation. So, each life becomes a lesson designed to grow the real you, the soul. And, under this concept, if you consider having millions of lifetime experiences, then experiencing one or more lifetimes as an amputee (or rich person, poor person, woman, man etc.) may simply serve to provide a perspective as you gain wisdom and understanding. For me, reincarnation is the only thing that helps to make sense of all this.
Follow up question/answer may be, to what end are we trying to improve our souls? Consider that each soul comes from the exact same thing (God, Creation, source, whatever you want to call it), and that our goal in being temporarily separated from the source is to learn, experience and grow through countless life experiences and incarnations. With the ultimate goal to rejoin the original source, only to do it again and again. It's a giant wheel of life, only, what we experience with this body is a very small part if the big picture.
So talking about being an amputee, being healthy, being rich, or poor, or tall, short, white, black, man, or woman for a given lifetime is simply too narrow a discussion. It's like trying to understand a movie, by analyzing a single 5 second scene. It can't be done.
Welcome to the forum!
The "Why won't God heal amputees?" question is really aimed at people who think that God answers prayers. If you are proposing a reality where every eternal soul has to exist in this universe at some point during their eternal journey, and in this universe there is no healing from "God, Creation, source, whatever you want to call it", this question isn't aimed at you. You have maneuvered your way into a less-contradictory position. Congratulations. Another good thing is that a proposal like yours, on its own, doesn't promote ignorant, unwarranted hatred that needs to be fought against.
But, your pretty-much harmless proposal just doesn't seem likely to me, even if I have to grant you that there is an infinitely-miniscule possibility that you made a lucky guess and got it right. There is a world of endless possibility.
I exist for whatever reason. I don't have free will. Free will is an illusion. So I'm here for the ride. On this ride, I live my life as though I have free will. I don't remember being born, let alone remember any existence I might have had before I was born. It's possible that I may only exist once. Even if there is reincarnation, from where I stand, it's possible that in every new existence, I will only be aware of my current existence, that is if I am even at least as intelligent and aware as a human is.
If there is an intelligent being behind my existence, I am definitely left in the dark for no reason that I can understand.
Many people like to assume that there is a "greater good" that we will all someday understand. The fact that we seem to be living in a universe that is indifferent to the happiness and suffering of conscious creatures, I calculate just as much of a chance as there being a "greater bad" as there would be a "greater good", but I would bet on a "greater indifference" if any.
For me, "I don't know" is definitely a better answer than "Therefore, reincarnation".
Once again, welcome. It's fun to discuss and learn.