This is a complete redefinition of division, because the natural numbers can be understood in a very explicit way in which the prime numbers cannot: the natural numbers can be understood as a counting. That is to say, the number six is not just the sixth natural number, it can be thought of as the sum of a collection, such as saying it is six objects or six ones. The fact that the natural numbers do not only form a sequence but also form a way of counting items gives a natural way of understanding the addition, subtraction, multiplication, and division operations.
But the sequence of prime numbers is only well-understood as that: a sequence. The second prime number is not two first prime numbers: it is just the prime number that comes after the first.
Because the second prime number cannot be thought of as two first prime numbers, redefining the operation of division to operate on prime numbers in the way you have just doesn't make any sense. Or, at least, it doesn't make sense in the usual way that division makes sense. Perhaps there is some new way to make sense of the operation, but I see no purpose in bothering with this unless some new way of making sense of this operation is found, which I really really doubt can happen.
Why must it be good for something? Binary is good for computers because it is easy to build computing devices which can store information in two states (on or off). It's also good for formal logic (where the states are valid/invalid or true/false, depending upon the context).
Not every conceivable operation you can do with numbers is good for something.