If we grant for the sake of argument that these are the facts, it's still not clear to me that banning the sale is a good thing.

Suppose we have three people, Alice (who lives a comfortable upper-middle class life), Bob (currently starving), and Carol (rich and in need of a kidney). Alice doesn't particularly care about money, but if there is a lack of kidneys she is willing to give one to feel that she is doing good in the world. Bob cares very much about money---if he can't sell the kidney he will starve to death.

In this hypothetical there are enough donors to fill demand even with a $0 price ceiling, so Carol doesn't care either way. But by banning the sale, the benefit of the transaction now goes to Alice (who gets some extra warm fuzzies) rather than Bob, even though Bob is in much more dire need.