If I shut off the water supply to an outdoor faucet during cold weather, is there much benefit to also covering the faucet using a styrofoam faucet cover or similar? My guess would be no, but maybe the pipes conduct heat well enough that it would still be worth it.
Metal pipes conduct heat very well.
You have another question indicating that you have been insulating pipes.
If a pipe is insulated from the warmth in the house and has a turned off hose bibb sticking out into -10F weather, it's far more likely to freeze if that hose bibb is not insulated from the -10F weather.
In any case, given all your other posts about insulating the house, even if the pipes don't freeze here's a heat leak that's easy to fix that you'll be paying for even if it does not freeze the pipes.
Don't insulate - use better valves
Change the faucet out to a different type called a "hydrant" aka "sillcock".
There is a long shaft going the length of the assembly. The spigot knob turns the actual valve at the other end. This means the length of the assembly does not have water in it (if you follow the rules!) when the valve is shut.
So make sure you convert to this valve type if you don't have it already. Note the distinctive arrangement with the spigot handle dead square-on to the wall, and vacuum breaker above.