No. The parent suggests having 2 machines (1 at each end), but each trading order would be issued by only 1 machine (not 2). So there is no synchronization issue.

Example: machine A instantaneously become aware of some event at exchange A. It sends an order to remote exchange B (which takes 5ms) and at the same time sends a matching order to exchange A. Total time = 5ms, which is the same as having a machine in the middle having a latency of 2x2.5 = 5ms. [Edit #1: I edited this sentence to reflect the fact the algo sends the matching order immediately to the local exchange.]

So I still don't understand why is there any advantage in having a computer in the middle.

Edit #2: you wrote an edit ("This way the two machines don't need to sync up to determine if one leg was done and one leg was hung. All the information is in the same program on the same machine so there is no distributed state to reconcile.") which I think finally clarifies the advantage in having a computer in the middle. You don't run the risk of the 2 computers at each end executing orders that conflict with each other.