How to estimate the average time complexity of greatest common divisor?
Since $n\geq x$ in your sum, $O(\log\min(x,n))$ simplifies to $O(\log x)$. Being very careless with constants here, but $$\frac{1}{n}\sum\limits_{x=1}^n f(x)=\frac{1}{n}\left(n\cdot O(\log n)\right)=O(\log n)$$

I'm not sure what this is calculating though; average runtime of all input sizes less than $n$?