How can the synthesized a priori be benefical to the world's operations?

Answer 1

It cannot.
Kant made synthetic a priori up with no real evidence.
Read Quine for a severe assault on the concept.

Answer 2

Let me think...

You're right.

An ampliative truth so powerful as to be necessary doesn't serve any practical purpose.

It's just like my knowledge of Campbell soup varieties. Necessary and contingent, no difference really for us people.

^^