Suspect they were IFR. All your points stand. First time flying things with a jet engine, I was shocked how much more fuel gets burned at low altitude. It almost always works out better to max climb to altitude and descend than to fly low and level. On a small jet, things can get spicy fast when ATC route you around at 5000' for 15 minutes or so. Three aborted landings would gobble gas like crazy.
§ 91.167 Fuel requirements for flight in IFR conditions.
(a) No person may operate a civil aircraft in IFR conditions unless it carries enough fuel (considering weather reports and forecasts and weather conditions) to—
(1) Complete the flight to the first airport of intended landing;
(2) Except as provided in paragraph (b) of this section, fly from that airport to the alternate airport; and
(3) Fly after that for 45 minutes at normal cruising speed
They were most definitely IFR. Not because of the weather but because IFR is required above certain altitude 18,000 ft in the U.S. and typically lower in Europe (depends on a country). Jets including small private jets are almost always on IFR. Airliners with passengers - always.
Why does it burn fuel so fast?
My guess is higher air density means more wind resistance, which acts as negative forward acceleration.
Not just that. Jet engines are efficient at higher speeds because the exhaust of the jet engine is fast.
If the plane is going fast as well, that exhaust is more or less stationary relative to the ground. The engine works to exchange the position of the plane with the position of the air in front of it.
If the plane is going slow, it's accelerating the air backwards. That's where the work is going, making the engine less efficient.
Think about it this way: if the jet airplane is tied to the ground, its engines are running at 0% efficiency, working hard to blow the air backwards. You wouldn't want to stand behind a jet engine when the plane is about to take off, when that's effectively the case.
The same applies to propeller-driven planes, of course. But those can vary the prop speed as well as propeller pitch, having more control on how fast the air is being pushed backwards. This allows the engine to be efficient at a wider ranger of speeds, particularly, at the slower range.
But the propeller has a limit of how fast it can push the air back. When the prop blades start reaching the speed of sound, weird shit starts happening [1]. So propeller-driven aircraft have a limit on speeds at which they can go efficiently.
Jet engines (turbofans when it comes to airliners) trade off low efficiency at low speed / low altitude (where the airplane is spending a small percentage of flight time) for higher efficiency at high speed / high altitude.
Variable pitch turbine fans[2] aim to address this tradeoff, but the tech has yet to catch on.
[1] https://en.wikipedia.org/wiki/Republic_XF-84H_Thunderscreech
[2] https://en.wikipedia.org/wiki/Variable_pitch_fan
That sounds like Oberth effect in rocketry, where the faster you go the more efficient your rocket be: https://en.wikipedia.org/wiki/Oberth_effect
they have nothing to do with each other.
I think about it like this:
Jet needs to suck air from front. If air is stopped, sucking is hard. If air is already being thrown at you, you don't even need to suck, just let it come in.
You are right that accelerating the air backwards more reduces efficiency but I think it should be mentioned that the jet engine has to accelerate the air backwards to do any work pushing the plane forward. Picking it up and setting it back down affects the air with a net force of zero and therefore the force pushing the plane forward is also zero.
So perhaps the differential air speed between the intake and exhaust is a big factor in the efficiency equation? The bigger the difference the more work is needed..
Variable pitch turbine fans sound very interesting! Perhaps in the future as tech improves and fuel efficiency incentives continue to increase.
So, newton's first law?