What I think your assumption is that the connection between the client and server at either end would be faster, making the local lag for that person better. But in reality the information about the other player's actions would still need to be transmitted, processed, and then displayed to you (local processing and rendering). Adding another server in between would just make this slower by having two servers process the information.
Client -> Server -> Client
Client -> Server -> Server -> Client
Which one is longer? Even if the connection between the two servers is as fast as possible, you still have the processing time that would have to exist to keep the servers in sync. (Not that any game server software is designed that way) Multiple servers is generally reserved for MMOs in a cluster, where the load can be dynamically shifted to optimize performance as players enter or leave areas or additional players join the instance. On the small scale it just wouldn't work. (Not to mention that you would be at the mercy of each individual server's access to the internet)
Typical propagation delay between any two points on a network is increased by roughly 10ns for every silicon device required. 23 ping is 23ms, so if you take out the distance between two points on earth, the remainder is the processing and signal propagation delay. Adding a whole server means going in via the Internet, through the router, through the network interface, through the chipset, to the CPU, processing, back to the chipset, through the network interface, back out through the router and back to the Internet, to repeat with your second server. No matter how it is done, it will add that much time minimum to your ping.