GPS is just a stand-in for the system they would use on Mars, which would be a much simplified version using existing satellites and the transport vehicle for the lander itself.
The atmosphere in Mars is fine for a quadcopter if designed correctly. How do you think the parachutes on NASA landers work?
I'm not sure what you mean by "simpler" system, what would it be? AFAIK it takes signals from 4 GPS satellites to get a fix, I think that it could be done with 3 having knowledge of the approximate position. That's with 3 or 4 satellites with GPS electronics in view at once. This implies that there need to be several more than 3 satellites in the constellation to be certain of having 3 or 4 in the correct position at any time. I don't think that the orbits of multi purpose satellites would be the ones needed for GPS.
Parachutes are used to slow the descent rate of the landers and are then cut away as the descent rate is still pretty high. Other methods need to be used to bring the lander to touchdown. I guess some engineers experienced in origami might be able to design a multiple rotor copter that would be compact in transit yet unfold properly once the parachute slows the descent rate etc, etc, etc.