The disjoint between what there should be and what there is - a hand but none - is what allows you to cast spells. And if you want to become a truly powerful wizard, both hands have to come off. Nor is it possible to replace the hands with prostheses, as anything taking the hand's place would interfere with the magic.
This means that great wizards are simultaneously very powerful and quite helpless. They can single-handedly blow up a castle, but they have to be fed by their servants.
Technically, everyone can learn magic - though there are certainly some people with a lot more aptitude, such as with all skills. Since it's not really possible to determine this aptitude without chopping off someone's hand, there's quite a lot of failed magicians around. Most of these are absorbed into administrative roles within magic schools, but some of them, embittered, go forth and pretend to have magic powers, hoping no one calls them on their bluff.
Of course, there's also a tendency to take off the non-dominant hand of thieves as punishment, which means there's also some left-handed thieves pretending to be wizards, and left-handed wizards assumed to be thieves...
Finally, this is all in the mind: it's not that there is a magic emitter in the human wrist that only needs that pesky hand out of the way to work. It's the psychological effect of losing a hand which allows for the magic to work. This means that there could be one or two magicians around with perfectly attached, functioning hands, but a delusion that causes them to think they're gone...