The mistake that is likely being made by Harris, as well as many others in western culture, is failure to take into account the events around which the Quran was being taught. Also, please forgive me, I haven't watched the video you posted yet but I will when I get time later today.
During the time the Quran was being spoken by Muhammed (it wasn't written down for a very long time- originally an oral teaching) was a time of upheaval. Followers of Muhammed were being persecuted by Judaism as well as the surrounding people who were neither Christian nor Jew (the "Pagans"). Muhammed had formed peace treaties with some, and in particular the Meccans. The Meccans, however, broke this treatise time and again, attacking Muslims while on their pilgrimages and otherwise just in general. Regarding this specific issue, you find verses such as this:
"They ask you, Prophet, about crescent moons. Say, 'They show the times appointed for people, and for the pilgrimage.' Goodness does not consist of entering houses by the back door; the truly good person is one who is mindful of God. So enter your houses by the main doors and be mindful of God so that you may prosper. Fight in God's cause against those who fight you, but do not overstep the limits: God does not love those who overstep the limits. Kill them wherever you encounter them, and drive them out from where they drove you out, for persecution is more serious that killing. Do not fight them at the Sacred Mosque unless they fight you there. If they do fight you, kill them - this is what such disbelievers deserve- but if they stop, then God is most forgiving and merciful." 2:189-192
There are many who take these verses out of context to say the Quran is proscribing violence. However, if one is willing to read it honestly, it is clearly not the case. What is being proscribed is self defense, and highly restrictive self defense so as not to overstep limits, I.e., be unnecessarily violent. The issue at hand was the persecuting Meccans were attacking Muslims while on pilgrammage. Pilgrammage was a holy time, and the Muslims weren't fighting back and getting killed or taken captive from fear of muddying their service to God. So these rules were given proscribing restrictive self defense, where they could only kill if necessary (others are trying to kill them), and they were to stop immediately if the assailants ceased their oppression AND BE FORGIVING as God is merciful.
The Quran does not proscribe these vengeful actions by terrorists because under no circumstance is a Muslim supposed to be on the offense. To do so is condemned in the Quran as a sin which, if left unrepented, will land one in hell whether Muslim or no.
