I would suggest that you take the time to reflect on the fact that if almost everyone disagree with you, it might point to something that you misunderstand. Either about the meaning of what they say or your understanding of the subject.

Their paper shows that the algorithm takes a constant time to compute the Cth coef. of the Nth step, for a given C and N. Not that the algorithm time is independent of C and N. C and N are not the input of the GCD. The algorithm running time is constant for the input of the GCD. It is dependent on C and N.