Given two environments that are identical, except for air density (e.g. Cape Canaveral, but at Mount Everest's height), would launching a rocket require more or less fuel at the lower air density?
[Physics] Does launching a rocket in lower density air require more or less fuel
airliftrocket-science
Related Solutions
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.
A 747 - can get you to around 35,000 feet. Still very much within the atmosphere.
So what do you do then? Launching a rocket from that point still requires an awful lot of kit, so while you have reduced your propellant requirements a little, the 747 still has to carry a launch platform, so you're not really getting anything out of this.
New technologies, such as that used by Virgin Galactic is managing to make this work, hopefully, with a hybrid model that does fly up to around 50,000 feet before launching the spacecraft section, but this is very new.
So the simple answer is - it used to require vertical rocket launches, and all the associated paraphernalia, but modern technology is moving towards fully reusable methods such as this.
Best Answer
At the start of the launch, the rocket has the largest mass of its entire flight. Any rocket that can make it to orbit necessarily is fairly big, making its fully loaded mass enormous. The combination of large size and large mass makes its relative air drag smaller than compared to a smaller and less massive rocket.
The rocket's speed is also a consideration. At maximum acceleration, the rocket becomes supersonic only after it has reached the very upper limits of the troposphere. This means it only start moving really fast after it has climbed above the most dense parts of the atmosphere. Since air resistance depends quadratically on speed, but the air density drops roughly exponentially with altitude, air resistance is hardly a consideration at all (for large rockets).
Naturally, if there is no air at all, there is no air resistance, so less propellant would be required in all. But with regard to fuel efficiency: for any rocket that can make it to orbit, removing the whole atmosphere would be less effective than launching that rocket from the top of Mt. Everest :)