In addition to other helpful suggestions....
1) As Noel pointed out, the actual temperature at the A/C thermostat may be lower than your lowest mechanical stepping and therefore will not start the compressor or fan.
2) The indoor fan will not necessarily run if the A/C is in cooling mode and is set in "auto" position, AND the T-stat setppoint is higher than the actual indoor temp.
3) Your extension cord will likely not provide enough current to over come the needs of the starting CAPACITOR for the compressor. My guess is that the "buzz" you hear is also "buzzing" when you run your generator, but you just can't hear that over the generator noise.
4) You A/C compressor may not try to start if the unit is low on refrigerant gas. A low pressure switch will prevent start. A knowledgable mechanic needs to check for a proper charge.
5) I suppose the obvious ( and most wishful solution) would be to carefully inspect ALL the rooftop A/C wiring. Small control wire connectors can easily be pulled off their connection, which could have easily happened when the shroud was broken.
Hope you find a simple and inexpensive fix