I reckon that if somebody were to cross reference the high achieving September/October babies that then fetch up in top universities, a very significant proportion would be children born to teachers, as it's the best time to maximise maternity leave entitlement.
Just by the fact of being born to a teacher, they are immediately in a position of statistically being more likely to attain higher standards due to the educational level of their parents, the financial advantage, the degree of engagement in learning, familiarity with educational language, learning and communication, etc, etc. And when you have parents in a position to plan a baby's birth date to that degree, you're getting people who are more 'sorted' rather than chaotic.
And yes, there is also the element of confidence from being the bigger, stronger, further ahead children in class which doesn't end at 11 - after all, even at 6th form, they'll be the ones able to drive earlier, get part time jobs for over 18s with higher wages sooner, go on holiday alone sooner, etc. They're used to being ahead, so they're often inherently more self assured because it's what they've always known.
But most of all, I'll bet they do better because they're more likely to be from a comfortable, settled, secure family background.
I'd aim for an autumn baby when first trying and then reassess whether this is still a priority if it doesn't happen quickly - I conceived twice almost the moment I'd thought about it, but then had infertility that was never solved. By the time I'd realised that it wasn't going to happen again, it didn't matter about advantages, it was hoping for a miracle that never came. After all, it's only another 3 months.