In the name of God, the Most Gracious, the Most Merciful.
Well, the reason we have to do certain things three times, is because God says so. Why does He say so? According to Sheikh Nezaam bin Yahya Ali, he says the following in regards to this question:
In Islam, accomplishing certain tasks in odd numbers is significant. Imam Tirmidhi mentions a Hadith in which Nabi (SAW) is reported to have said, “Certainly Allah is odd and loves that which is odd.” In Arabic, the first (odd) plural number is number three. If a person counts from number one, the very first number which denotes multitude is number three, which is odd. It is for this reason, that in certain aspects of Shari'ah, when a large amount is intended, then it must be done in the least odd number which denotes multitude, and that is number three.
So, no, it's not a coincidence. There's no such thing as "coincidence" in Islam. Everything happens for a reason, but we humans, don't always know that "reason".
As for you being agnostic, here's my e-mail if you want to ask me some questions personally, because you being agnostic means that you don't know whether God exists or not. I can give you proof that He indeed exists.
I hope that answers it, and God knows best.
May God reward you, peace be unto you.