As a libertarian, I support that in principle. But let's look at it in practice.
The airlines could create a consortium to take over the air traffic control function and then buy all the airports. The cost of air travel would skyrocket, and there would be a lot of airline bankruptcies, but as Joseph Schumpeter has said, that's just "creative destruction". Only those who could afford to fly would do so -- until the costs were amortized.
We could end the highway subsidy by installing a transponder in each automobile. Once a month, the motorist would get a bill for his travel on local and county roads, which would remove the need for a property tax for local and county road-building. He would get a bill from the state for his travel on state-maintained roads, which would permit repeal of the gasoline tax and other car-related taxes. And he would be billed for his travel on the former interstate highways, now sold to private entrepreneurs, that would permit the end of federal gasoline taxes. But the motorist would raise holy hell, and FReepers would start muttering about "the Communists trying to track our movements and herd us all into the cities to control us."
Which level playing field works best? A totally private transportation infrastructure, or a mix?