Why does differential signaling send complementary signals instead of just pairing the input and ground voltage?
"Ground" as a concept needs some clarification. If you have just one signal line and one ground, then yes, it's hard to tell the difference. But if there's anything else going on, it matters.

All AC signalling involves current flow, even if it's being measured as a voltage on the receiver. At a minimum, you have to charge/discharge the parasitic capacitance of the receiver, and the signal wire will have a capacitance to ground as well. Note that the common mode behavior only holds if the two wires have the same length and are a constant distance from each other.

So if you have a signal wire and a ground wire, then the current in the signal wire must be matched by a corresponding current in the other direction in the ground wire. If you have lots of signals, the ground wire will contain a mashed-together copy of all the signals. Therefore it's advantageous for each signal to have its own ground. If you look at VGA, you'll notice that each signal gets its own ground because of this. If you look at 80 pin IDE, each pair of signals has a ground wire between it in the ribbon cable. Those are to prevent the signal wires inducing currents in each other ("crosstalk").

Once you've accepted that each signal must have its own matched ground, it's more natural to embrace the two as a matched pair, disconnect one from ground and connect the two of them together via a termination resistor network, and drive / read them as a differential signal.