In my scenario, a machine knows the local exchange price as of 0ms ago (instantaneously) but knows the remote exchange's price as of 5ms ago.

You are saying it is better to know exchange A's price as of 2.5ms ago, and exchange B's price also as of 2.5ms.

But why? Either way you are ignoring 5ms's worth of trading activity. Either 2.5ms on both exchanges, or 5ms on one exchange.

Edit: I think I understand the advantage - see my edit #2 in my post above.

You do: that's the signal that's sent over in 5ms. You then combine it with the local price 5ms ago, and you can construct the exact same instruction the computer in the center would have come up with 2.5ms ago, based on the price in both locations 5ms ago.

Both sides can this way make identical decisions to eachother, at the same time, and identical to those that would have been made by a computer in the center.