Great question and answers.
I don't know if I'll ever shoot a wedding (I'd be happy to for family or very close friends...no charge...free meal). Wedding photography is the big leagues, in my opinion.
Also, I shoot center point and recompose. Obviously this works better with a little distance between camera and subject, as you can lose focus on your subject (if the subject is on the same plane as the initial focal point then it doesn't matter. Closeups are very difficult for keeping on plane).
Since I don't shoot weddings or sports, a 6d is good.
But the OP shoots weddings. If that's the case, and if you can find the extra money, then why not get what works best? In other words, are there any advantages the 6D possesses over the 5D3 other than price?