None of the above! Though we suspect at least a few sculptors breathed a sigh of relief when they first found out they didn't have to carve hair.
The dismembered statues are the result of a surprisingly practical move by the Romans. They were meant to be taken apart and put back together again in different configurations, like ghostly Cronenbergian Legos. The bodies were purposely generic, so detachable heads could be paired with the torso of your liking. Say, for example, you needed to whip up a cool new goddess on the spot.
Luis Garcia/Wiki Commons
Also great for sticking your head on, theme-park-cutout style.
This was most handy in the political realm. Whenever a new ruler came to the throne, the Senate could decree a damnatio memoriae. This meant that the old emperor's memory had to be erased from history. They'd remove his likeness from everything they saw, the same way you delete old Facebook photos of your ex. The old heads would be thrown away or recycled. Then, when the new emperor began his reign, a fresh batch o' noggins was sculpted and mounted to the franken-leader statues. And if the new emperor ever got too cocky, the Senate could clear their throats and point to the throat seam. He'd get the message.