They do but you have to already own the shares or find a borrow. Otherwise how are you going to sell the shares to your counterparty?
If you create a naked short, what does your counterparty actually own when he buys the shares from you per the contract? The shares don't exist! So what does he own? What he would own is a claim to force you to buy them and give them to him, but that's not what he's supposed to get per the put contract. He's supposed to get real shares, and in exchange he gives you the strike money.
I agree, it's total BS. They should allow naked shorting in these circumstances to ensure that the spirit of the put contract is preserved.