It sounds like the drivers would be likely to receive a large number of routes that would need to be displayed. Therefore, I understand your concern about graphing them all together. However, you can make this look good and not so overwhelming. For example, matching start and end points, like your A1, B1, A2, B2 idea, using bright/bold/dark colors, and then drawing the line or route in between using a lighter tint of the color to avoid cluttering the view.
This still presents two questions: Would they rather just see lines in between vs. routes? And do they need to see all of the routes at once on the same graph?
First, would they rather see lines in between vs. routes? My problem with lines or arcs is that it shows a route that the drivers can't take. People know they can't drive thought blocks and houses, so seeing a line that doesn't follow the streets is confusing and misleading. However, like you said, they don't need to see the directions, and they could obviously take a different route between a start and end point (although they are likely to take the fastest, so showing that route would probably be accurate most of the time). I think showing the directions allows the drivers to best visualize each route. I really like how the Google maps app presents their alternate routes - it uses the lighter colors like I mentioned earlier to draw the focus to the selected route and the routes are named using one major difference in the directions.
Next, do they need to see all of the routes at once on the same graph? What is the purpose of seeing all of the routes? Does it allow them to optimize their route? Although I doubt that they can pick multiple people up on the way or choose the order that they will be completing the rides. Are they picking their next route out of ones in the area? If that is the case, then I think you would want to highlight ones whose starting point is closest to the driver's current location. If the driver has to complete all of these routes in a given order, then I don't think there is any reason to show them all on one graph. Show them on separate screens in order with next/previous navigation buttons (I am leaning towards just having a next button so they can hit it when they're done with their current route, but I could see if they are curious about their next route). Also include the number of the current screen, like 2/18, if the purpose of showing them all on one map is so they can get an idea of how many they need to complete. I don't think there is any reason to have this on a timed carousel. Let them peruse at their own speed or as they are finishing one and going on to the next. If this is an idea to make it hands-free, then I don't think a driver should be looking at the screen for the number of routes * 2 seconds while they're driving.