A head server has a lot of roles. Their primary job is as the manager of your server team, and ensuring that they are doing their job properly. The deal with birthdays/anniversaries/etc. They are also responsible for dealing with any food allergies, and ensuring that the food quality is meeting with the guests expectations. They also play an important role in finding out if there are any problems with a guest's trip - not just their dining. Since they have the opportunity to just chat with the guests, they often learn about issues in other areas, and can pass that along to the right people to fix it. To be honest, if you don't have special needs, there isn't much a head server can do to screw anything up. (The Hawaii head server dug himself a major hole, and then kept digging - not easy to do.)
But a good head server is something much more. We've had several good servers over the years, who have gone out of their way to make our trip more magical. They have a huge box of pixie dust to pull from, and really have the ability to make your trip something special. Here are a couple of specific examples:
1. On one cruise, we knew a family traveling where one of the kids got sick. Mom stayed behind with the sick child in the room, while Dad took the other child to dinner. Head server learned about it, sent a runner down to the room with that night's menu, and then had dinner delivered for Mom, and some soup for the sick child.
2. On night 2 of a longer cruise, a request was made to the head server for a special order desert (Key Lime Pie). The head server made it happen, and the pie arrived at the table the next night for dinner. He then asked us what we wanted the following night. We laughingly told him to surprise us. And for the rest of the cruise, a special custom made desert was delivered each night for the table to share. We still talk about the strawberry tarts that we had never seen before, and have never seen since.
3. On the Fantasy Maiden, just after boarding, we ran into a head server we've known for a few years and were chatting with him. He asked where we were sitting. We told him, and he quickly looked it up in the computer. He shook his head, said 'No, that won't do', and pulled out a piece of paper and wrote down a new table number for us. (We learned later he moved us to a stronger server team). That night at dinner he introduced us to our server team as his 'family.' You can imagine the message that gave to the servers.
I take your point about not tipping the restaurant manager at home, but you also wouldn't tip the busser or bartender who the server got the drinks from. But in many cases, they are tipped - by your server. The server splits the tips with those people (or at least the smart/good ones do). When cruising, they just call the roles out separately for specific tipping, based on their individual service. And think of the total tip amount. ~$6 a night for the three positions, per person. That's less than I would tip at say Chili's when having an appetizer, entree and dessert.