But, yes, the Quill is very much a part of canon and I see very little way of getting around it. In fact, having a character not find out their a wizard until their teens or older is a plot point in a lot of Mary Sue fics, so I generally don't recommend it just on that principle, unless you have a very, VERY strong explaination for it.
How is it a Mary Sue trait if the character is weak? The later your magic shows, the weaker it is, right? So if a character shows magic after their eleventh birthday, how would that make them a Mary Sue?

They wouldn't start Hogwarts with year three only because they are thirteen, but probably with thirteen in year one... I imagine that extremely discomforting...

I really don't see any Mary Sue potential, but actually a very interesting plot idea.

That being said, it would be quite impossible for someone to begin attending Hogwarts AFTER they had turned eleven just because they hadn't yet shown any signs of having magic.
I just don't get why that would be. Do you really think that the headmaster would deny students the opportunity to learn and receive a proper education just because they are a couple of years older than the others? I really can't imagine how that would go... That child obviously has magic and needs to train it (and especially needs the school to maybe pepper up those weak magic cells), but the school shuts its doors in the child's face...

We have heard of students that had to repeat a year before (like Flint, I think) or certain classes (Crabbe and Goyle), so they would have been older than the average too.