> If traders have to be covered with the actual stock, would that mean they can only short Foo shares up to the amount they own? Surely that's equivalent to just selling your holdings?

I had selling a call option in mind.

If you own stock, you’re loaning it out to short sellers all the time. Almost any brokerage includes in its contracts the right to loan out its customer’s interest to shorts.

The short seller has an account with a brokerage, the broker is ultimately on the hook for making sure that any borrowed short interest is returned. This is why brokers have margin requirements.