What purpose does Zoro's third sword actually serve? Zoro has trained with Santoryu for basically his entire life, and has completely perfected it by this point. But, just from a fighting perspective, it would be quite difficult to get enough leverage/support with a sword in your teeth to actually do effective damage to anything (at least, without Haki). Unless this is just a classic case of "this is a fantasy world with fruits that give people magical powers, so it's possible for Zoro to have a strong enough neck to use a third sword effectively". 

Most of the times we see him fighting, it's usually only with his two swords in his hands, and we barely ever actually see his third sword do anything. We often see implied strikes after he finished Oni Giri or other big finishing moves. 

It seems Nitoryu would be just as effective as Santoryu in most situations.  What exactly does his third sword do in a normal fight, that is so important for his style to hinge on it? Did the third sword become more useful in normal combat after the timeskip (not including the use of Haki, because if that was included, obviously it would be more useful), or is it still basically the same?
Tl;DR Zoro's Third sword serves a critical and crucial purpose in his ability to use the "Three Swords Style" which is exponentially more powerful. The synergy between the multiple swords is what makes his sword style more complete and usable in an effective form. The difference lies in "between using (just) three swords and three sword style".

Detailed: For us to understand the use/purpose of Zoro's Three Swords you've to assume that One Piece exists in a separate universe and our understanding of things are not the same. This simple assumption allows us to remove a bias when trying to explain "Fantasy Universe Physics"

From a young age Zoro has been shown to be shown possessing unnatural strength, capable of besting opponents far beyond his age. He used Two Swords because he assumed, two were better than one. However, being bested in a thousand duels to Kuina made him rethink. Post her death he decided to use her sword (Wado ichimonji) as one of his primary weapons, just to fulfill both their dreams of being the strongest swordsmen. To this purpose he trained his mouth and neck to extreme lengths to attain the strength, accuracy and flexibility required to use the sword.

Now since no one else has been shown to use Santoryuu in OP universe, I think we can safely assume Zoro invented and developed the style and its techniques while growing up. This chapter front page shows how his juniors are trying to imitate and learn his style in his old Dojo Source: Chapter 807: Cover page 
 

Now, what makes this viable his Zoro's effort and training to develop a comprehensive "Three Swords Style" aka Santoryuu. He is not using just Three swords, he is using the "synergy" between the three swords to gain more power and use techniques which are impossible to use with one or two swords. This was explained early on by Zoro himself in his battle against the NyanNyan brothers. 

We've seen numerous instances of this throughout One Piece. Zoro talking about his (in)ability to fight with one sword against Nyan Brothers, his surprise when Mihawk blocks his attack or him mastering new moves. (Pound Cannon, Asura etc.)