It's a quite bad analogy because the googolth digit of pi is completely constructive. (You don't need to calculate it, but it is constructive)

P = NP proof could be not constructive.

My intuition was that if a proof for P = NP exists, it would be incomparable to the kind of Pi example you provide- Pi is defined as an irrational ratio, so the existence of whether x digit of Pi exists. It would instead be like saying, 'the x digit of P is 7, and here is a proof that is not a straight calculation'. The idea of a proof which can demonstrate knowledge of X digit of Pi, without verification, doesn't click for me.