This question is specifically related to
The $gcd(x,y)$ is solved in $O(T(n) log n)$, where $n$ is the number of bits of $x,y$, and $T(n)$ is the time to multiply two $n$ bit numbers, as described in
so when it reads "if you're dealing with big integers, you must account for the fact that the modulus operations within each iteration don't have a constant cost. Roughly speaking, the total asymptotic runtime is going to be $n^2$ times a polylogarithmic factor. Something like n^2 lg(n) 2^O(log*n). "
Isn't this the same thing as stating "the fastest $gcd(x,y)$ is solved in n^2 lg(n) 2^O(log* n)"? If so, then
How could a total asymptotic runtime exceed the upper bound of an algorithm's runtime?
If the $gcd(x,y)$ is solved in $O(T(n) log n)$, shouldn't this be true no matter how large $n$ is?