Ultimately, the goal is to cure people, where the definition of “cure” is that the patient ends up happy and healthy. In terms of health, there’s no difference between the body of a woman and the body of a man. A healthiest that a woman can ever be isn’t any more nor less healthy than the healthiest a man can be. Consequently, there’s no difference between a cure where the patient ends up taking the body of a different gender. Functionally, they’re at 100% physically and, ideally, they are now at 100% mentally.
Amputation, by definition, leaves the person less healthy physically. A cure which accedes to the demands is impairing the person.
Beyond the physical health of the patient, at the end of everything, there’s the question of “are they going to stay satisfied”? If you turn a man into a woman and she’s satisfied as a woman and will continue to be so, then there’s no large loss. If you amputate a person’s left foot and the next day they want their right foot gone, then you amputate that and they want their right hand gone, etc. then there’s no knowing whether there will be a point of satisfaction or if the person is going to keep continually maiming themselves further. You may find out that the “cure” you have given them is not a cure and subsequently, all you have accomplished is to create a person who is just as poorly off as they had been before, mentally, and now missing a limb.
All of that said, gender reassignment is not a perfect process, in practice. The result (particularly if the person was beyond puberty at the point that the process began) is not someone who perfectly resembles a member of the opposite sex. And in either case, the person will be functionally “sterile”, since the change is largely cosmetic at this moment. So it’s a bit of a misnomer to say that the person ends up as a 100% healthy member of the opposite sex, given the limitations of today’s technology.
Further, there’s the question of how well they are accepted by society as a true member of the opposite sex. It is presumably going to be hard for a person to think of herself as a woman if the majority of those she interacts with do not view her as a woman, and instead view her as a man or as an odd or undesirable “third” gender.
And of course, restoring a person to their original gender will encounter the same problem. The process (particularly for men) is not completely reversible. So if a person is discovered to have not been satisfied by the transformation and wants to go back or wants to become something else (e.g., a cat), then suddenly you’ve run into the same problem as we fear with the serial amputater.
At the end of the day, what matters is the average end result based on actual case histories. Regardless of whether two things might, in some fantasy philosophical model, be the same thing, if in the real world, entirely different approaches work out, then so be it. And that’s where we are on this particular topic. I believe that the evidence is in favor of gender reassignment and against amputations, and consequently, all imperfections of either approach aside, what works in practice is what works. But certainly, there’s a lot of progress that needs to happen before we can really say that we have a cure in place.