I am picturing a mathematically inexperienced programmer writing a (pseudo)random bit-generating routine that calculated the millionth digit of pi and returned it.

I'd be extremely impressed if a mathematically inexperienced programmer could pull of a program that calculated the millionth digit of pi!

Could they justify their code by pointing out that they don't know what the millionth digit of pi is, and so they can treat it as a random bit?

I say yes (assuming they only plan on treating it as a random bit once!)