When the government pays for things, it distorts the economics, because the true cost of that thing is being hidden from the consumers. By including the cost of enforcement in the cost of the service itself, the market can balance better.
For example, when allowing horses, that means the plane uses additional fuel. Usually the cost of the fuel is included in the ticket, so as fuel becomes scarcer, prices go up, and less people fly. But when the government starts paying for some of that fuel, it will hide that price increase from consumers, so they will fly more than they should, and use fuel faster than they should.
Also, almost 90% of Americans have flown[1], so it's not like filers are some tiny minority.
[1] http://survey.airlines.org/