Assalama aliekum
I would like to interject something that I have noticed here in Kuwait (among Kuwaiti youths).
The children DO take Islamic studies from at least grade 1, if not before (it is actually REQUIRED by law for all Muslim students to take Islamic studies starting in grade 1 - before that is optional based on the parents' choice of schools).
However, there is MUCH emphasis on memorizing the Qur'an - very LITTLE emphasis on understanding the meaning. Yes, they learn the words, but the understanding is sorely lacking. And this is not even to the point that there is no rhyme or reason for the order that is presented to the students in the government Islamic studies program. About a year ago, there was a big to-do over the fact that in one of the new grade 1 books, the verses they had for the students to learn were battle and marriage based. The argument was that there were other verses that would be more beneficial to the youngest students - they didn't understand the marriage issues or the history behind the battles anyway, so why have them memorize those first?
My high school students - when they were studied for their exams most recently (end of January), I asked them about the Qur'an verses they had to learn for the exam (they would be required to write them out completely on the exam paper). I asked them to recite the verses to me - they did; then I asked them what one of the verses meant - in my class of 14 high school boys, the best they could come up with was "it is something to do with charity". I asked them, what do you mean - don't you learn the meaning with the verse? They said, No, they are only required to memorize it - they spend the entire class period just repeating repeating repeating. Mind you, these are ALL native Arabic speakers - they understand the individual words, but the MEANING is different (think if I said, "Wow, that is cool" in English - native English speakers immediately know that I am not talking about temperature - an English language learner may understand the words individually, but may not get the concept).
Honestly, memorization without understanding is not productive - it may be a first step, but in the Kuwaiti curriculum, there is not much required beyond memorization (they also memorize hadiths without interpretations or understanding).
So yes, it is possible to read and memorize the Qur'an without understanding it, even by native Arabic speakers.
Lana