But that's not a complete model, as the output will start changing the moment the input changes. And a short burst will not appear so on the output.

It will change instantaneously, but with a magnitude that decays exponentially with distance from the place the current is injected. The way signal propagation works is that you have "active" currents to ground that react to voltage changes in a nonlinear way. These lead to wave-like behavior in the transmission line, though it is quite nonlinear and harder to model than a straight up inductor.