This doesn't really explain why those velocity variations occur in the first place, or am I missing something?

It sounds like "We observe velocity variations on the wing and these correspond to pressure variations that create lift due to conservation of energy." But it leaves the question on what is causing the velocity variations in the first place.

One other weird thing needed if you want your momentum numbers to balance: every plane in the sky is pulling the earth up towards it ever so slightly.

If you just mean there's a gravitational force on the Earth from the plane then, sure, but that's true for any object (including you and me). For an object on the ground, the normal force of that object on the ground balances their gravitational pull on the Earth, so the Earth experiences a net force of zero from the object and so isn't accelerated by it.

But I guess you mean that the plane has a net force pulling the earth towards it. But that would violate conservation of momentum – in fact the net force is zero, just as for an object on the ground. The plane's wings push the air down (the reaction to that is what keeps the plane up) and the resulting downdraft of that air exerts a force downwards on the earth.

That's all assuming the plane is in steady flight. If it's taking off, or just ascending, then overall it is pushing the Earth away from it. Conversely, when descending the Earth is pulled, overall, slightly towards it. The same thing happens when you jump: you push the Earth away from you (a much smaller distance than you travel away from it!), then on the way back down the Earth travels back towards you.

Crud I think you're right. I had it in my head that in steady flight, the plane had zero change in momentum, whereas, the air, collectively, gained net downward momentum. So to balance it all out, the earth must be gaining upward momentum (though of course spread over such an enormous mass as to make the velocity term imperceptible.)