If you have a lock with no keys, it's a non-starter; the door has no purpose. You'd be better off sealing the entrance with bricks.
If you offer a website with an API then CORS is an enhancement in that you're helping protect users from getting owned or hijacked. Alternatively if you can't stomach the risk, you could take down your API, which is about as useful as solving a math equation by multiplying both sides by zero.