The problem with this "just right" longer-path answer is that it's not based on physics.
So while it sometimes gives the right answer, it fails more often than it succeeds. It's like computing 16/64 by cancelling the 6's i.e.
16/64 = 1
6/ 64 = 1/4
Granted, it gives the right answer in this particular case, but it's not based on mathematics, and if you're doing arithmetic this way you're going to arrive at the wrong answer more often than not.
Likewise, if you try to understand lift without a clear notion of the definition of lift, you wind up making nonsense statements like having a foil that "generates lift on one side and not both sides". This is impossible, unless you are working with a different definition of the word "lift" than that used by scientists and engineers.
I think I'm done here. For those lurking who've gotten this far in the thread, read the links I've posted for more info, and don't believe the non-physical misconceptions stemming from the longer-path and equal-transit-time fallacies. They're simply wrong.
The (simplified, and good enough for most people) reason why a foil generates lift is that the fluid is deflected by the foil. Fluid has mass, so if the fluid goes in one direction, the foil is pushed in the other direction in accordance with the law of conservation of momentum. The more fluid you deflect ( either by increasing the angle of attack, increasing the size of the foil or changing its shape, or by finding faster moving fluid to deflect) the more force you get from the foil.
The rest is just details.
Granted, this idea does not explain why some foils are more efficient than others - that requires a more detailed analysis, and for that you need real physics, not the nonsense expoused by the discredited 'popular explanation' based on longer paths and equal transit time.
Well its not altogether wrong the way the Newtonian model can trivially be shown to be. BTW the Babinsky explanation is not exactly misleading but it is so incomplete that it really might as well be considered wrong. If you want we can go into what it is that he assumes and tries to say and ends up being misleading on - but Id rather do that in another post.
Where the "longer path" model is not altogether wrong (and Babinsky does get this part wrong) is that in the case of sails, even a sail with no mast (ie a lufftape only jib like on many a modern skiff ) has a stagnant air pocket on the windward side of the jib, and this pocket serves to create a "shorter path" for the "bottom" stream vs the lifting stream
Secondly, what is really doing the work - as you allude to but don't quite get to the root of - is the curvature. Yes the curvature induces centripetal accel - but that's not really the bit that causes the flow and pressure changes we observe when instrumenting the leeward side of a sail. And this is where the real explanation gets very hard to explain to folks - What's causing the accelerated flow (and it does accelerate as even your video shows) is that you essentially are trying to force more air molecules through a smaller space.
HUH? But there is essentially infinite space to leeward of the sail you say.
Sure but that also means there is essentially infinte mass resisiting displacement and compression. in the lateral direction.
So what you really have are the same flow dynamics you would have if you took a foil and pulled it through the center of a pipe. the walls of the pipe act as constraints on the lateral displacement of the fluid. now the fluid has to go somewhere (imagine a tube that is closed with a track down the middle and the foil on the track - and it is all filled with water. I can clearly move the foil through the water so the water has to go somewhere) and in the case of the upper/lifting surface, the ONLY way it can move ffrom the LE to the TE in sufficient volume to (hey FS this is for you - To comply with the law of Conservation of Momentum) is to move more incompressible molecules through the same space faster. That means that as a molecule moves across the curved surface, since it is doing so faster, it "vibrates" (brownian vibration is what causes pressure of any sort) against the surface fewer times. Thus there is lower pressure there.
Now in the Real World of open air, the essentially infinite incompressible mass of air acts as the walls of the pipe. And yes there is some lateral compression of air, but that has to do with viscosity efffects. So it is the laterall displacement of some of the sttreamflows into the spatial location of the other stream flows that forces them to accel.
And as they accel. they cause a drop in pressure.
Since the lateral accel of a curved surface is greater than of a flat surface (where the lateral displacement is limited to the AoA, the flow velocity over a curved surface is greater and hence there is even lower pressure than for a flat surface. Thus you do get some push from the newtonian deflection, but you also get "suck" from the reduced pressure on top. and that is instrumetably measureable.
So in essence, because the air molecule is being moved further (travelling a longer distance) in the curved surface case, it generates more lift than the flat plate case. So the "longer path" model, while inaccurate in its completeness - is less wrong than the hash that Babinsky puts forth (which while not actually wrong, is so incomplete it might as well be wrong).
The "vortex sheet" is one way of modeling this because it allows you to take into account the differing accel and decel patterns we actually measure on the surface of the foil. so it is a more amenable calculation model for DiffEqs of the real world.