Is it a problem or a red flag when a facial plastic surgeon does not have admitting privileges at a hospital?

When you have admitting privileges, you often have to take call for the emergency room.  This care is often unreimbursed-think slave labor.  To avoid this, some surgeons decide that they do not need admitting privileges. Honestly most patients get admitted and followed by a hospitalist anyway.  Admissions after surgery are very rare.  Lack of admitting privilege does not mean that a patient can't be transferred to a hospital.  It is a bit of a red herring.  However, I would also ask if the doctor has the appropriate medical malpractice insurance to the procedures they do or if they are "self-insured."  Also determine if they are in good standing with their State medical board.

Plastic surgeons who perform surgery generally have admitting privileges at a local hospital, and is probably routine across the country. The hospital allows overnight stay and increased level of medical care before or after plastic surgery, if appropriate. In addition, accredited surgical centers require their surgeons to have admitting privileges at a hospital. Best,Dr. Chaboki

If you have not found any evidence of adverse actions it is likely he/she just dropped admitting privileges.  Malpractice insurance coverage often requires the surgeon to have privileges at a local hospital so the privileges may be other than admitting.
Some sort of privileges are important in case there is an issue that requires admission to the hospital.

Absolutely, I would be concerned.  If a surgeon does not have admitting privileges at a hospital, I would run the other way.  There are plenty of great surgeons in this country.  Explore other options.