That chopper was stripped to the bone so that one dude could fly in it, with a fuel time of 1 hour. It also landed on the summit which is more flat. Don't think it would be easily and safely able to carry an addition (dead) body that's probably perched on an incline, let alone carry rescuers. It takes upwards of 6 hours to chip the dead body off the ice (if it's been frozen there for a while)

The rotor would have to be enormous and fast, and the helicopter incredibly light with very little payload. One or both of those constraints are probably beyond our materials knowledge, particularly a controllable rotor that would work.

Passenger jets are fine up there because they haul ass, often north of Mach 0.7-0.8, and push as much of the thin air as possible over the wing. (One of the cool things about commercial air travel is that most people don’t realize they’re flying at transonic speeds, but you’ll occasionally see evidence of it in flight.) It doesn’t take much slowing down for a passenger jet to start descending, and ultimately stall, at cruising altitude. The flight envelope at Everest altitude is much smaller than you’d expect.

For a helicopter, you’d need to replicate that set of circumstances, but with wings rotating above you instead. One has to worry about other conditions, too: in the '70s, when Boulet set the helicopter altitude record above 40,000, the engine flamed out on account of how cold it was and wouldn’t restart. High altitude helicopters are a very difficult design problem, and there isn’t much economic demand to solve it.