How are "preferred routes" assigned if there is more demand than capacity on a route?
Air traffic services (at least in the US) are handled on a first come first serve basis. That's the overarching protocol. Beyond that, it's situation dependent.

In the air, controllers give pilots what they want if possible. In practice the situation is quite fluid. Maybe all three can't go on the same path now, but in a minute there will be enough clear airspace to make it happen. Or maybe they go on a route very close to but not exactly what they requested.

If (for some reason like a prominent jet stream) many airplanes are filing the same (or very similar) flight plans so that certain airspace will be overwhelmed, there is an entire division of the FAA dedicated to deciding where and when aircraft need to be rerouted. Here's some info @mins linked to in a comment.

Also, your question seems to assume competing airlines. For all ATC cares, A, B, and C could all be flights of the same airline.