Bird respiratory systems are more complicated than those of mammals.
The lung itself is a compact organ with air channels running through it, and a system of air sacs draws air through these channels, always in the same direction, whether the bird is breathing in or out. The blood vessels in the lung run parallel to the air channels, with the blood flowing in the opposite direction to the air. This counter-current arrangement allows a bird to extract oxygen from the air, even when the air pressure is too low for the dead-end lungs of mammals to operate.
Bar-headed geese migrate over the Himalayas, doing much better than mountaineers (mammals!) struggling below.
Actually 2,700 m is not all that high. Mountaineers function all right at that height, and plenty of birds (especially waders) fly higher when they migrate, even without any mountains to get over. We do not have a usable theory to calculate the actual rate at which a bird can extract oxygen for the air at a given height, but we hope to get a handle on this by observing the maximum rate of climb that our geese can manage at different heights, as they climb up to get over the Greenland ice cap.
Geese are indeed well insulated, but when they fly the thinly-feathered area under the wing is exposed to the air flow. That is necessary, because the problem in flapping flight is disposing of excess heat, rather than keeping warm.