Q:

Why is the wrong explanation of "air travels a longer distance and creates a lift" so popular?

When I was learning for my license, one of the first diagrams I remember was about the wing profile. The air going around the wing and on the upper side it has to travel a longer way, thus generating lower pressure and bang, plane is flying.
Same explanation already back at school.
See my other question: if the theory was right, why can planes fly inverted?
So here's the follow up: why is this wrong theory so popular and still part of books?
Wouldn't it make sense to teach students how a wing really works? I mean just look at any RC plane meeting - you'll be amazed what weird designs are capable of flying if there's enough engine power. 

A:

Long story short is that both Newton's third law and the Bernoulli's effect are two different ways of explaining the same phenomena. Accurate equations have been written to explain both. Newton's third law, the air being forced down and forcing the airplane up, is actually the effect of lift though not the cause. The cause of lift is the change in pressure.
The falsehood that is taught is usually that the two little air particles actually meet up at the same point downstream at the trailing edge of the wing at the same time. This is completely false. In fact, if you were to actually track two air particles, one over the top, and one across the bottom, the one going over the top of the wing will probably arrive much sooner than the one on the bottom! [Even though the distance IS longer] It picks up that much speed. Also they don't actually really meetup again afterwards at all.
The real pivotal relation (which I learned in my Advanced Aerodynamics class at ERAU using John. D. Anderson's book Intro to Flight) is the Kutta-Joukowsky theorem. Taking the time to explain this to a Private Pilot student would probably be a waste of your time and theirs but I will sum it up here anyway to be fair to the question. I will assume incompressible flow because after about Mach .3< it starts to get more complicated.
The Kutta-Joukowsky theorem takes the integral of velocity times the cosine of the incremental angle of the distance along the closed curve. This is a quantity called circulation. The lift equation is then formed as lift equals density (altitude basically) times velocity times circulation. So if you really want get in-depth here is an article from MIT that explains potential flow theory. 
What you will find is that the cause of lift is indeed directly related to Bernoulli's principle in that the change in pressure is what creates the lift. However it has nothing to do with particles meeting up at the trailing edge because they never really do meet up. This doesn't discount Newton's third law either: lift can be explained that way but it is the effect of lift, not the actual cause. And yes, like you mentioned, even a flat-plate for example can create lift when angled right in a wind tunnel. 

A:

Wouldn't it make sense to teach students how a wing really works?
No, not really.
Ask any (good) CFI and they'll tell you that there are certain topics that a student really needs to understand properly.  Examples:

What's a stabilized approach and how do you fly one?  (A disappointing number of students tell me it means holding the same airspeed all the way down)
Why do we clear the prop area?  (You'd think this would be too simple to screw up, but there's been more students than I've got fingers on my throttle hand who haven't yelled "clear!" until they're already cranking the engine.)

And so on.
How a wing really works is not one of these topics - the consequences of misunderstanding a stabilized approach are that you screw it up and your landings are terrible (or even dangerous).  The consequences of misunderstanding prop safety are that someone loses a hand or dies.  The consequences of misunderstanding how a wing generates lift are...well...the wing keeps flying anyway.
Now, if the student is an aerospace engineer, they'll need to know the real story, but for a pilot, they can continue in their ignorance for their entire career and never be negatively affected (unless they meet an aerospace engineer in a bar and get into a fight).
To extend the point, I myself tell students that to properly understand GPS, you need to get into the theory of relativity, but for a functional but incorrect understanding, you can think of it as "DME-in-space".  It's very wrong on several levels, but it's enough to satisfy the needs of that category of student - same as the wing pressure-differential explanation.

