Saying (2n)! ≈ 2^(2n)*(n!)^2 means he's approximating the central binomial coefficient (2n choose n) as 2^(2n). In reality, 2^(2n) is really the sum of all binomial coefficients (2n choose k) for k=0..2n. So clearly this is going to be an underestimate, as we're approximating the sum of 2n terms by only one of them.

However, since the biggest term in the sum of 2n positive numbers is clearly within a factor 2n of the entire sum, the ratio of the logarithms is at least log(2^(2n) / 2n) / log(2^(2n)) = 1-log(2n)/(2n) which converges to one pretty quickly.

And, in fact, a tighter analysis would show that the error decreases faster than what I outlined.

The fact that Sterling’s approximation is only an approximation