This question is specifically related to
https://stackoverflow.com/questions/3980416/time-complexity-of-euclids-algorithm
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
How is the Time Complexity of a Function Problem different than its corresponding Decision Problem?
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?