Air Bus?
We've already got low-fare airlines dispensing with reserved seats, meals, etc. on many flights. What about the ultimate in no-frills air transport, the airborne equivalent of a city bus?
Instead of buying a ticket for a particular flight on a particular day, you buy a coupon for any flight between a given pair of cities. The coupon isn't tied to a given date or time (though it might have an expiration date): you simply arrive at the airport, and get on the next flight departing for your destination city. Coupons could be sold in ATM-style kiosks at the airport, eliminating the need to plan ahead, or in 10-packs for frequent travelers.
This would require that flights depart often enough that there's a flight departing shortly after whenever you get to the airport. Departures would have to be at least every hour for most of the day (but probably not more than every half-hour). Many pairs of cities already have flights at least hourly; for example Minneapolis and Chicago, or New York and Washington.
Advantages
No more missed flights or connections. If you get to the airport late, just get on the next flight.
No more arriving early for security and then waiting around. If you get to the airport early, you can catch an earlier flight.
No more overbooking or getting bumped. If there are too many people waiting and the airplane fills up, there's another flight soon.
No pressure to be at the airport at a particular time. Arrive when it is convenient, and step onto a plane.
The airline can dispense with the expense of tracking reservations and seat assignments.
Everyone understands the fare structure
But wait....we can make this even simpler, by pricing all the flight coupons the same. Short flights cost one coupon, medium flights cost two coupons, and long flights (coast-to-coast) cost three or four. Now you're not only freed from having to decide your departure time, but also your itinerary. If you're a business traveler, you don't have to plan your trips in advance at all. Just go to the airport with a fistful of coupons. Jump on the plane in L.A. to go to a meeting in San Francisco. When the meeting in SF is done, your boss tells you that she's scheduled a presentation in Denver, so you jump on a flight to Denver. And so forth.
Or, let's suppose you had planned to go to Chicago, but when you get to the airport, you realize there's a big snowstorm there and no flights are getting in or out. Instead of sitting at the airport, you phone a prospect in St. Louis and ask to move your meeting up a few days. Jump on the flight to St. Louis. You can go to Chicago the next day if the weather has improved.
This also lets people change their routes on the fly in response to crowded flights, mechanical problems, or bad weather. Going from Chicago to Denver, but a flight got cancelled because of a mechanical problem? Catch a plane to St. Louis, and from there to Denver instead. With bus-style air service, any city with connections to at least two cities can become a hub (in other words, the flights form a mesh network instead of a hierarchical network). This might even make it more profitable to offer air service to smaller cities, by driving traffic through those cities to change planes.
Even More Advantages
Passengers would route themselves around flight delays and cancellations. No more rebooking.
Airlines could respond to congestion in one part of the country by adding flights to parallel destinations. Snowstorm in Chicago? Cancel the flights to Chicago, and fly the planes to St. Louis and Detroit instead, and let passengers change planes there. This is much more robust than the centralized hub model.
Of course, there are disadvantages, too. The biggest one is figuring out how to handle checked baggage. Since the airline wouldn't know which flights you'd be traveling on, it can't make sure your bags are in the belly of that particular plane. The easiest thing to do is have the passenger mark the bags with the ultimate destination, and let the airline handle the route--but the TSA would have fits over that. The airline can run a sort of parallel package delivery service, where you have to pay for every checked bag (maybe through bag coupons, just like flight coupons), which would encourage passengers to limit themselves to carry-on.
The other disadvantage is that not all destinations will warrant frequent flights to or from anywhere. Even flying regional jets, a metro area would probably have to have at least a million people to justify hourly air service. There are 49 metro areas in the U.S. with at least a million people (many of which are probably underserved for air service today), which is enough to build a pretty good network, but there are some states which wouldn't get served at all, like North and South Dakota.