This argument apparently has some intuitive appeal. There is something of a problem with this argument. Consider the counter-example of the memorization of the canon of Scripture.
It's quite common in Christian schools (I have not done any formal study of this, but I believe it is quite common) for children to memorize the canon list of all the books of the Bible. Thus, numerous children can recite the books of the Bible, beginning from Genesis and ending at Revelation, sixty-six books in all.
This fact, however, does not mean that there have never been any canon questions or canon disputes. The practice of having children memorize the canon list is not something that started with the apostles. Thus, we cannot argue that it would have been impossible for anyone to add or subtract a book, because "every child knows that there are sixty-six books, and that they are called Genesis ... Revelation."
The same problem exists for Qur'an memorization. In order for this argument to have weight, someone needs to show that there was an early, widespread practice of memorizing the entire Qur'an.
On the contrary, the records we have suggest that the early memorizers of the Qur'an did not memorize it in its entirety, but rather in parts. For example:
'Casualties were heavy among the Qurra' of the Qur'an (i.e. those who knew the Quran by heart) on the day of the Battle of Yalmama, and I am afraid that more heavy casualties may take place among the Qurra' on other battlefields, whereby a large part of the Qur'an may be lost.(context and citation can be found here).
If there were many people who had the Qur'an entirely memorized in that time, it would seem impossible that "a large part of the Qur'an may be lost." Therefore, assuming the testimony we quoted above is accurate, there were not many (if any) people who had the Qur'an entirely memorized in that time. Thus, the argument from Koranic memorization fails.