In the comments, David Marcus asked a really important clarifying question, one that I hear often from elected officials in low-density places:
Could the empty-running coverage-oriented buses be replaced with some sort of dial-a-ride system running full?
Yes, where you have a low-density and therefore low-ridership market, you can often replace empty fixed route buses with Dial-a-Ride systems. Dial-a-Ride in this context means a form of demand-responsive service, usually run with small buses or vans, where the route followed through its territory is different on each trip based on who calls and asks for the service. Dial-a-Rides can be ordered by phone or website, just like taxis but with less flexibility. Pickup times may not be ideal, and unlike a taxi your Dial-a-Ride van will make other stops to carry other passengers along the way.
But Dial-a-Ride systems rarely have many more riders than the fixed routes they replaced. More commonly, these transitions increase productivity (riders/unit of service cost) by acting mainly on the bottom of that ratio, i.e. they serve the same small numbers of people with service that's less expensive to operate.
In a typical big urban region, the highest-performing all-day routes -- the ones that run high frequency through dense parts of the city -- usually score over 30 boardings per revenue hour and often closer to 50. (A revenue hour is one transit vehicle operating for one hour.) Physically, it's almost impossible for a Dial-a-Ride style of operation to serve more than 10 boardings per revenue hour. Now if you have a low-density area where empty fixed bus routes are running around scoring 5 boardings per revenue hour, I may be able to replace those buses by half as many Dial-a-Ride vans serving basically the same people, and thus scoring 10 boardings per revenue hour. That's an improvement, but it doesn't begin to bring these low-density places into the performance league of high-density ones, where 20 boardings per revenue hour is considered disappointing.
In other words, Dial-a-Ride can be a great option for optimizing Coverage service, but we're still talking about Coverage service in either case. Dial-a-Ride can't raise a low-density area's productivity into a league that would match service in high-density areas. Again, the very nature of Dial-a-Ride caps its productivity at around 10 boardings/hour, because that's about the maximum number of times in an hour that a van can get to a specific address or street-corner where someone has called for it. (Remember that when talking about low density we're usually also talking about labyrinthine street patterns that are designed to make it hard to get quickly in and out.)
I should clarify, too, that when I say Dial-a-Rides usually don't attract a lot of new riders, I'm talking narrowly about a case where a bunch of empty-bus fixed routes are replaced by Dial-a-Ride vans. I'm also talking about Dial-a-Ride in the transit context, where multiple passengers are consolidated into one vehicle by making multiple stops, and thus often following a circuitous path. There are higher-end forms of Dial-a-Ride, such as door-to-door airport shuttles, but these are another matter.
UPDATE: Anonymouse kicks off with a good, concise statement of a common comment:
It seems like dial-a-ride systems could benefit massively from the improved dispatching that computers make possible, and things like smartphones could greatly improve the user interface as well. You could have an app that lets you order a ride from "here" (determined by GPS) to a point that you pick on a map, letting you know how long it will be until you are picked up and how long the trip will likely take, and how much it will cost. Very different, and likely to be much more popular, than the earlier model of "schedule your ride at least 24 hours in advance".
Dial-a-Ride is an area of great technological ferment. There are lots of opportunities to improve the ways that Dial-a-Ride can be dispatched, making more spontaneous use possible. These improvements could vastly expand the demand for Dial-a-Ride, but they don't change the physical limits of supply.
Look at it from the perspective of the driver, who sees the calls pop up on the screen. Regardless of the dispatch method, Dial-a-Ride means each passenger requires her own bit of routing and a separate stop. In this style of operation, will you really be able to pick up a passenger every three minutes, on average, all day? That's what it means to achieve 20 boardings per revenue hour, an astronomical performance for a Dial-a-Ride but still very low for an urban fixed route. The only way to get higher performance is if you get lots of people traveling as groups, i.e. not requiring the individual responsiveness that is the whole point of Dial-a-Ride.
This whole issue is important because it's common to hear talk of Dial-a-Ride innovation as though it somehow changes the fundamentals of service to low-density areas. It does not. Labor remains the dominant element of transit cost in developed countries, so ridership per revenue hour remains a good unit for assessing performance. Dial-a-Rides can do wonderful things, but they're still Coverage services, and in any large urban area, you wouldn't run them if maximum ridership were your only goal.