Id get a second opinion. The tech notes indicate that a mode door is needed and does not point to a AC auto amp at all. I'm pretty sure Nissan doesn't sell the motors separate from the whole heater box, so the quote is confusing. The box is probably at least $800 plus maybe 8-10 hours of labor, so I'm not sure where that number came from. The notes also don't address the lack of temp change, which is different door with a different motor.
These motors are installed on the side of the HVAC box. If you pull the trim panels off and get down on the floor, you should be able to hear and feel with your hand, these motors moving things as you change the settings. Usually what I find is that the motors are working, but the plastic hex on the box side is stripped out, so they aren't actually controlling anything. It sounds like that's what the tech was doing on the mode door motor, but found that the motor wasn't working, but had what it needed to do so. No mention of the other door motor though, so who knows.
These motors are installed on the side of the HVAC box. If you pull the trim panels off and get down on the floor, you should be able to hear and feel with your hand, these motors moving things as you change the settings. Usually what I find is that the motors are working, but the plastic hex on the box side is stripped out, so they aren't actually controlling anything. It sounds like that's what the tech was doing on the mode door motor, but found that the motor wasn't working, but had what it needed to do so. No mention of the other door motor though, so who knows.