Why don't swamp coolers push air instead of pulling air?
The main reason swamp coolers are typically arranged that way is to minimize their footprint. That arrangement allows the motor to be inside the air-handler box, instead of poking out the end.

Additionally, moisture is not as much a problem for electric motors as heat. The overwhelming majority of the moisture passing by the electric motor is vapor, or "humidity" rather than mist. If there's enough mist to make the motor wet, then the cooler isn't working properly because evaporation is the key ingredient. Most electric motors fail because the insulation around the wires that make up the magnetic coils melts or deteriorates. This shorts out the coil, which no longer generates a sufficient magnetic field to drive the motor. Keeping the motor cool would actually be a more important consideration than keeping it dry. But even the moisture available to the motor shouldn't be getting it wet.

I can't think of any reason that reversing the fan arrangement couldn't work.