Alright, we'll go with your analogy.
You're in a big city. Each morning, you look out and see that the roads are packed with cars that are barely moving. Everyone wants to drive to work in a car: so many people, that only a few can, and those few still fill up the lanes completely. So, you have to settle for biking to work every day.
If you stubbornly threw away your bike and got in a car, you would never get to work on time. If everyone got into their cars, there would be traffic jams everywhere, and nobody who drives would ever get to work, except for a lucky few who were close to the front when the jams started.
If Riot let you wait for just one role, then you would never get to play if you chose the most popular ones. It's not just that you have to wait a long time. As more and more people join and get put into queue for that role, an infinite line will build up; there just isn't enough of a supply of support players to fill the demand. The only resolutions are either having a lot of mid players giving up after waiting, or kicking everyone out of the queue after twenty million mids are waiting for supports and the matchmaking servers crash.