

The “120 volt cable”, assuming you mean NM-B aka Romex, is rated for up to 600 volts if you look at it closely. It is absolutely acceptable to use that wiring for a 240V circuit, as long as you wrap colored (not green) electrical tape around the white neutral wire to indicate it’s another hot.
Yes, there are 3-conductor (plus ground) wires one can also use for switches and 240V circuits with neutral. That neutral can be used to have 120V and 240V together (your oven may use 240V coils, but the light bulb probably runs on 120). Doesn’t mean you need to have it, if your 240V circuit doesn’t need a neutral. My air compressor is just a motor that can run at 240, no neutral needed, and its outlet is wired up with the same kind of Romex used for a 120 right next to it (with black tape to indicate a second hot)
Here’s the fun part: while you’re all talking about their ice cream, technically it’s not legal to call their product that. You won’t see it anywhere on the menu. I think it has to do with the milk/cream/egg/sugar amounts? There may not be egg at all, but can’t remember for sure.
Anyway, all you’ll ever see on the menu is “soft serve”