A short summary of the paper mentioned in another answer and another good site.
Basically planes fly because they push enough air downwards and receive an upwards lift thanks to Newton's third law.
They do so in a variety of manners, but the most significant contributions are:
- The angle of attack of the wings, which uses drag to push the air down. This is typical during take off (think of airplanes going upwards with the nose up) and landing (flaps). This is also how planes fly upside down.
- The asymmetrical shape of the wings that directs the air passing over them downwards instead of straight behind. This allows planes to fly level to the ground without having a permanent angle on the wings.
Explanations showing a wing profile without an angle of attack are incorrect. Airplane wings are attached at an angle so they push the air down, and the airfoil shape lets them do so efficiently and in a stable configuration.
This incidence means that even when the airplane is at zero degrees, the wing is still at the 5 or 10 degree angle.
-- What is the most common degree for the angle of attack in 747's, 757's, and 767's
![right](https://i.stack.imgur.com/zH0zq.png)
Any object with an angle of attack in a moving fluid, such as a flat plate, a building, or the deck of a bridge, will generate an aerodynamic force (called lift) perpendicular to the flow. Airfoils are more efficient lifting shapes, able to generate more lift (up to a point), and to generate lift with less drag.
--Airfoil
In your own question you recognize that the Bernoulli equation is the wrong thing to apply to this situation, because obviously there are dissipative losses involved.
My preferred way of looking at this is recognizing there is a lift to drag ratio that exists as a metric for aircraft. This can be 4:1 or 25:1 depending on the plane. Regardless, provided that we accept the existence of this ratio in the first place, then the airlines are justified in the claim that more weight $\rightarrow$ more fuel. Limiting the discussion to cruising, it then becomes a simple multiplication of weight times lift to drag ratio to find fuel use.
The other flaw in your argument is, of course, the assumption that speed can be increased to compensate for more weight. A cursory reading into the flow path of turbo-machinery will disprove this. The jet engines will be most efficient at the designed cruise speed and rotation speed, and any deviation from that will alter the angles at which the air hits the rows in the turbine, causing efficiency to decrease. In the real world, drag also tends to increase as some power of velocity, which in itself will probably predict some marked decrease in the lift to drag ratio, again, making the plane consume more fuel. If the plane uses different altitudes to compensate for different weights with the same velocity, then more dense air will obviously cause more drag. It's true that these are ultimately viscous losses, but this flow is turbulent, and its likely that drag will scale as something close to $\propto \rho v^2$ (density times velocity squared) as a result of that fact. As the density increases fuel consumption will too.
Best Answer
This is known as Ground Effect. Not to be confused with flaring, which is a technique used by pilots to gain lift by increasing the angle of attack as airspeed decreases.
Technicality, you can flare an aircraft at any altitude. The higher the altitude, the faster the airspeed of which you can flare an aircraft before stalling due to air thinning as altitude increases.