From a historical perspective, there is no denying that a real invasion (but not migration which is a big difference) took place. If the Berbers has not been recently converted, there may very well have never been an invasion as the Arabs had been extended pretty far at that point. The Berbers provided an addition of manpower. Iberia was ruled by dynasties and what they brought along with them.
To suggest that there was some type of super-rapid cultural diffusion would not only require turning a blind eye (I assume that ojo ciego would not translate) to what happened. Islam has been described as fighting its "way up population gradients" (quote McEvedy) in many circumstances but to my knowledge never came by a people reaching out and absorbing it across a border/frontier. That was not what occurred at all.
Iberia was a tempting target for an army which was very much based on the rapid acquisition of booty. The Visigoths, trying to exist as Arian Christian rulers of Catholic subjects, had never ingratiated themselves with the people as the Franks did after Clovis' choice of Catholicism.
They had also been involved is desultory wars with the Byzantines and had only recently kicked them out of the peninsula completely. The Visigothic Kingdom was not in a position nor may not have had the unity necessary to repel a significant assault. The Islamic rulers, being so far away from reinforcements/help, were likely less harsh than Muslim rulers in other places, especially after the Umayyads were left ruling Iberia and nothing else following the establishment of the Abbasid caliphate. (And the murder of most of their family) After they caliphate here fell apart into warring emirates, they still had no reason to make the Christian states in the north look attractive to their subjects. They probably did not have the manpower to suppress any potential peasants revolt.
They did bring learning and culture to Iberia, although aside from Algebra very much of it was gleaned from ancient Greeks and Persians. This environment was not pulled in by the people of Iberia. The magnitude of the invasions may very well have been overstated, but lets remember that this forum is about history, not changing history.