You can get your prescription from any pharmacy, but you can only send a prescription to one pharmacy at a time. So if you choose the local CVS and they are taking a long time to fill it, you have to transfer the prescription to another pharmacy (even if it is just a different CVS store).
Are you saying that if you have a prescription, you can go to any pharmacy without pre-arranging anything, and they will give it to you? Are all pharmacies linked to a central system so they all know your prescriptions available and whether they’ve been filled?