i think the whole not breathing myth has been confused and mixed up greatly. I believe its that vampires do not need oxygen or air, but they do breathe.
They do not need to breathe, they say this in both Buffy and Angel. They also have no pulse or heart beat so they couldn't cut off blood to the brain. They also technically shouldn't bleed since they have no blood moving through their veins, but they bleed several times throughout the series.