I think that the main issue here, and I haven't finished to watch the video, is that Von Sivers wants to link the Quran, the arab conquerors of the 7th century and the different and sometimes incoherent tales about the faith of the arab conquerors.
Events reported by externals sources (which have no real agenda as nothing is at stakes- God, religion whatsoever...)are incompatibles with what can be extracted as historical in the Quran as the text is a-historical and the 9th narratives.
If the Quran is against the christians and the arab conquerors come with the Quran, how come they side with Monophysites in Egypt, how come Mu'awiya go and pray on Golgotha,etc,etc.
A little part (the elite) of Arab conquerors comes (in the West) with Quranic texts (not necessary all), they side in Egypt with Monophysites. against Constantinople, Normal strategy.
Mu'awiya is not "Muslim" designation of today. Visiting Golgotha (after 640) does not mean that he is Christian. There is already a prayer building in the Temple Mount since 637 made by those in command who have came in the West, Mu'awiya never destroyed it. So he is aware that there is something, Is he an adherent of this prayer building? Maybe yes, maybe he is not already, as he have not real consistent informations about the "texts" (future "Quranic" ones) therefore of "Islam", as those who have the texts around him. They're all monotheists. He has let the prayer building being built. That poses no problem for him.Visiting Golgotha can be a respectful visit, either he does not know that the texts his friends have, denial the Crucifixion, either it is a respectful visit because all the Jerusalem population is Christian or both.