The SETI people making this busted argument are implicitly assuming p, the chance that life arises on a suitable world, is not "too small". And there is no basis for making that assumption.
The argument is generally that scale and probability favor the existence of intelligent life, assuming Earth is not somehow exceptional in the universe.
Assuming Earth is exceptional seems just as problematic.
Welcome to the Fermi paradox.
Why are you holding p fixed and allowing N to increase? You are implicitly assuming N is in that regime where N > 1/p. That's just assuming p > 1/N, and there's no good reason to make that assumption.
The comment that started this thread: "That's 10^22 opportunities to get it right. We can't possibly be alone."
That is not mathematically rigorous, but it is, from the Bayesian perspective, enough to move the needle.
That is obviously false; zero observations will never cause a Bayesian adjustment.