But it seems that the amp instead assumes the two grounds are the same, and just connects its ground to the RCA ground. That seems like a bad assumption to make to me.
No, that's how it's done. A common mantra around here is "tie ALL grounds together" because everything needs to know what +5v is in relation to (for example.) Same goes for audio. If you lift the ground of the RCA, you no longer have a circuit. It's just a spurious input with no reference.
Now, in your case, the RCA shield and the PSU ground come from the same place -- your power supply. So, it would in theory still work. Also, in theory, you do have a ground loop between the amp and Mighty. One path is from the RCA shield output on the Mighty to the amp input. The other is from the Mighty to the PSU to the amp power input. This is unavoidable, because if we assume the amp PSU ground is the same as the RCA shield ground, and don't tie them together at the amp, then what happens when you plug your amp in to a PSU, and feed audio from a battery-powered iPod instead of the Mighty? There's no path from the iPod ground to the amp.
If all your grounds are firm and low-impedance, there will be little potential voltage difference between the audio and PSU grounds, so the loop "shouldn't" cause a problem. What you're dealing with isn't necessarily a ground issue. These things can be tricky. I'm fighting a similar issue on a plate amp I built for a powered speaker with a microcontroller for inactivity power-off. Very faint high-pitched "bzzzzzzzzzzzz". Annoying.