But if the two terminal source is already balanced ... The balance is already there
Yes, but it is easily destroyed. As soon as you connect that source (say, a cart) to a receiver, everything goes out the window. The receiver ends up defining the mode of operation, whether single-ended or balanced.
Consider: The secondary of an output transformer and the primary of an input transformer.
Excellent example, Steve. This situation works best when only one of the center taps is grounded. That prevents ground loops via power cords, etc. In your example, you connect center taps, which is fine. You are correct that no current will flow in the center conductor. It can be disconnected and you still preserve the balance, no change in performance. However, this again ignores the big picture. What did we miss? That damn third terminal again. It is still there, although in a minimal fashion. The connection is via leakage. The insulation in the transformers and wiring is imperfect. But it exists. It may be 1000M ohm from primary to secondary, or to ground, yet it is there. It keeps the common mode voltage of the system in check. Sometimes. If the insulation is really good, then the cable can pick up static charge over time. Without any common mode connection, the coils float, and are at the whim of external corruptive influences. It is possible that the static charge build-up is so great that the voltage difference ends up breaking down the insulation, causing an arc. That is, unfortunately, the real world we live in. In this example, the leakage path is actually needed to keep the common mode voltage within reasonable limits.
Another good example is the telephone system. It appears at first glance to be a two-wire system. Tip and Ring. But that is only with blinders on. In the big picture, there is an Earth connection too. No currents are supposed to flow through Earth, but it does provide a handy common mode reference. It is only connected at the CO. Now if the balance between Tip and Earth and Ring and Earth gets unbalanced, guess what? We get currents through the Earth, which is not good. Hence, maintaining balance is again crucial for performance.
A Simple Alternative
This is a clever connection that accommodates a 3-wire system. But it's not really a balanced transmitter because there is no signal on pin 3. Unless! Unless you receive with a center-tapped transformer, which then back drives the cable so that the signals are indeed balanced. That is why this is so clever.
jh