It's impossible to judge whether something that exists now would exist absent the government. After all, the economy is so state controlled, especially in transportation, that nothing has really been put to the market test.
That being said, the current cost of flying might be too high (from taxes) or too low (from subsidies). But regardless of what the price would end up being, it's the fastest transportation we have and I'm sure that some people would be willing to pay for it.
However, flying in a full 747 gets about 100 miles per gallon per person. Not sure how expensive their particular fuel is, so I'm not sure how economically efficient of an option it is.
But in high school my friend's dad had a small plane, and he flew us across Nevada. It couldn't have been too expensive. They did okay but they certainly weren't rich. It's possible that air travel would continue in this form.