So to make sure I understand your argument, are you saying it is easier because the county does not have good, up-to-date information, or it is easier for no actual reason.
Because either way you are having to call, drive to the office, and in your case wait.
Not all private physicians have elected to carry the vaccine, and they certainly do not know when they will receive it.
Asking for them to post information online and having a plan is really not a big city service.