"Echado por el aire" sounds a little strange. "Salir despedido" normally means to gain great speed really fast, no matter what the origin of the impulse is and it doesn't describe what the stop will be either, even in some cases, like in accidents, when we assume that there is an abrupt crash afterwards.
I'd say "volar por los aires" or "salir volando" are synonymous expressions, depending on the context.
- En cuanto le dije que su madre lo buscaba, el niño salió despedido de la escuela.
...el niño salió volando de la escuela.
Here, it's the will of the boy to run towards his mother, and my exaggerated description of his running as fast as he could.
- Al chocar con el camión, el chofer del auto salió despedido del auto, porque no llevaba cinturón de seguridad.
...salió volando del auto...
Here, it's because of an impact that had nothing to do with the will of the car driver, and it's almost literally that he "flew" through the windshield and out of the car.
- En los helicópteros hay un mecanismo que te permite salir disparado, para salvarte en caso de accidente.
...salir volando del aparato, para...
Here, you know you can be ejected out of the helicopter, but assume the landing will be soft.
- Los juegos mecánicos tienen arneses y barras de seguridad para que no salgas despedido del asiento.
...para que no salgas volando...
And here, the bars and straps act against the inertia that would make you fly away from your seat, by keeping you speeding with the ride, rather than gaining a trajectory of your own.
The expression "salió despedido hasta el agua", in the original example, describes the fact that the car was already going so fast, that when it missed the road, it looked like flying ("salió volando") and crashed into the water.