Yeah, the problem with your sampling experiment (as your suspicious observation makes it sound like you’ve started to realize) is that this distribution has a very narrow tail to 1 that you won’t hit by accident but nonetheless contributes very significantly to the computation of the mean.

You correctly computed that the mean of log(foo(n)) is exactly −n, and the median of foo(n) indeed shrinks like e^−n (more specifically like e^(1/3 − n), I think), but the mean of foo(n) is exactly 2^−n.