Physicians want to be accessible only via specific channels, to protect their time and to ensure there is consistent process across patients. Most of the time as a patient you can’t call them directly, unless you work through the page operator or you plan ahead with them on predefined phone number and time.
Unless they're using their personal cell phone, this should be a basic feature of the hospitals phone system, it is a feature decades old. No reason the caller ID should be set to the DID of the doctor.
