In addition to the plane not being propelled forward, the reason a plane flies is the difference in pressure above and below a wing. Without air flowing around the wing, you get no pressure differential, resulting in no flight.
This is not really true. They teach this in basic PPL aerodynamics, but Bernoulli's principle is not the main reason why an airplane flies. On a standard C172 there is a pressure differential for sure, since the wing camber is different from the top and bottom, but on symmetrical wings this is less of the case. Also, if you fly a 172 inverted, it will still fly, but your AOA is much different to produce the same state of flight.
Anyways, the main reason why an airplane flies is Newton's third law. Even on something as small as a Cessna, over 13ft of air above the wing is being thrown down as the wing plows through the air. This downwash creates an opposite reaction which gives you lift. This is also why the treadmill doesn't work. Think of a wing like an oar in the water. The wing has to plow through the air to make lift. The wheels turning woudn't do anything. Even w/o an engine the airplane still produces lift, just not enough to counteract gravity!!! Well, gliders can cheat this, but you get the idea.
Again, this is just late night ramblings. There is a lot more to it if you guys are interested... in another forum for sure. Check Jetcareers.com... lots of technical stuff there to see. I just had to comment on the Bernoulli thing, since I got bashed by the FSDO on my CFI checkride for that.
Take care.