Why couldn't the physicians put the patient under during the procedure?
There are a couple reasons it's not preferred.
The procedure is not extremely invasive or very painful and can usually be tolerated awake. Most procedures that CAN be tolerated awake ARE done awake. General anesthesia has more risks than just anesthetic locally and an anxiolytic to relax the patient. It often requires airway protection (e.g. MAC or intubation) which also carries risks, and generally requires anesthesiology present in the room.
Sometimes if there is concern that the patient cannot handle it and would move excessively due to anxiety or such, then they may do moderate/general anesthesia on a case by case basis.
This is not something I could specifically find, but I can theorize that if there were to be disruption of an atherosclerotic plaque resulting in stroke, rare but a potential risk of the procedure, then the physician could identify signs of stroke far better when the patient is awake then when asleep. Interventions to treat stroke are time-sensitive ("time is brain" is the saying) so a delay could have serious results.