Your neighbor is largely correct, though you may be able to begin a selective program of pruning, cutting only sections of the hedge each year to 2 or 3 inches from the ground. Delaware Cooperative Extension recommends cutting all 4 year old wood to the ground. I've seen it recommended that spring is the time for rejuvenation of Privets. Often the recommendation is to do a third of the hedge, though this may seem arbitrary. Perhaps it is to not drag out the process of your neighbors recommendation longer than necessary?

Wayne