Why doesn't this RC circuit simulate as expected? (i.e. what am I doing wrong?)
Before starting the transient simulation, the simulator performs a DC analysis which concludes that the capacitor is fully charged so once the transient analysis starts, since your source doesn't change, the output doesn't move either.

The small movement you see in your first simulation waveform are probably due to a small numerical computation error difference between the DC and transient solvers. Once the transient solver "sees" an error it tries to correct for it in the subsequent time steps. Try zooming out your axes on the voltage and current to see how much those waveforms are actually moving by. My bet is that they are pegged at 9V and 0A respectively.