Just because you didn't get your particular solenoid working with whatever AC voltage and frequency and waveshape you applied, but did get it working with DC, again with unspecified voltage, is little proof of anything.
Solenoid coils are inductors, but also have significant effective series resistance. In any case, the magnetic field strength is proportional to the current thru the solenoid. When driving a solenoid with a fixed voltage, its impedance must therefore be taken into account. In the case of DC, in the steady state the inductive part of the impedance doesn't matter and the current is simply the applied voltage divided by the real part of the coil's impedance (the resistance). Many solenoids are specified to be used in exactly that way.
Some solenoids can be driven by AC. There are two issues with that. First, AC causes the magnetic field to flip twice per cycle. If the solenoid is driving something with fixed magnetism, that's not going to work. However, most solenoids simply employ magnetic material in the plunger without any permanent magnetism. In that case, the average of the absolute value of the magnetic field matters. Since the current is the voltage divided by the coil impedance, and in this case the inductive part of the impedance must be taken into account, a solenoid will require a higher AC voltage to generate the same force as a DC voltage.
In either case, there is a maximum voltage spec for every solenoid. This limit is generally due to internal heating. The resistive part of the impedance dissipates power proportional to the square of the current. At some point this is more heat than the coil can dissipate without getting too hot somewhere. The manufacturers will tell you what the voltage and current limits are, usually both for AC and DC cases. All you have to do is follow them.
I have used the H11F1 opto-isolator as a volume control. The output section is a JFET, the input is an LED driven by just a few milliamps. Use it to replace the mosfet to 'short' the signal or in series with the signal. If used to short the signal insert a 10K to 100K resistor to limit the signal current, and a 100 ohm to 1K resistor in series with the JFET drain pin (pin 6) to limit the short current. Use a 1K resistor on the output of the op-amp to limit drive current to the LED side (pins 1 and 2) of the H11F1.
I found this link to DIY projects using the H11F1 as a gate or gain control.
Best Answer
Internally it's taking the AC and converting it to DC. I can't say why, but it's almost certain that it's because they want to have both positive and negative rails, and it's easier to do that if you're getting AC off of the wall-wart.
I suspect that the design (if not the product) is over ten years old, though, because it's gotten so easy to use switching supplies to generate negative voltages, and the switching frequencies are so far beyond audio that the old problem (putting a ca. 20kHz "hum" into your audio) is just not an issue any more.
The only even semi-sensible reason I could see for it not being done is because an older-technology switching supply would be a Very Bad Idea in an audio circuit, and the audio market tends to be quite conservative in what they accept.