I should probably stay out of this, but I'd like to add a few points.

I don't think anybody will disagree that there is downwash directly behind a lifting wing of finite span. But to state that the momentum of this downwash is in some way equal to the lift glosses over one inconvenient fact: There is also UPwash outside of the tip vortices, and the net momentum of this upwash is equal and opposite to the downwash between the tip vortices. The net vertical momentum in the entire "Trefftz Plane" behind the wing is zero.

It's easy to see this if we examine the wing's wake one trailing vortex at a time. Each vortex has equal and opposite vertical velocities on the left and right, so the net vertical velocity integrated over the vertical plane is zero. Adding another vortex, and then another, and another, doesn't change this, since the velocities just linearly superimpose.

So we have to be more precise when stating something like: "The wing lift is equal to the downward momentum in the wake". Well.... sorta. Depends on how one defines "momentum in the wake". How much of the wake? Just behind the wing between the tip vortces, all the way up and down? Or the whole vertical plane behind the wing? You get a different answer for each choice, anything from the full wing lift to zero.

Here's the precise way to apply Newton's law for the airflow about a lifting wing:

First you must define a "control volume" of air which contains the wing. The size and shape of the volume doesn't matter, but you have to define a volume if you want to invoke Newton correctly.

One can define two vertical pressure forces acting on the air inside this volume:

1) The force F1 imparted by the wing from inside the volume, equal and opposite to the lift: F1 = -Lift (the lifting wing pushes down on the air).
2) The force F2 imparted by the air outside of the volume, via the unbalanced pressures on the volume's surface.

There is also a net vertical momentum change of the flow through the volume, such as the difference between inflow upwash and outflow downwash. Call this vertical momenum change delta(M).

Now... Newton sez (drumroll): F1 + F2 = delta(M)

All the squabbles here arise from the fact that F2 can be anything, since it depends on the shape and size of the chosen control volume. Some possible choices:

a) Volume is a huge cube with the wing in the center. In this case, F2 = Lift, so F1+F2 = 0, and delta(M) = 0.
The net momentum change of flow though the volume is zero. So the "no downwash" people are correct in this sense.

b) Volume is huge slab up and down, and fore and aft, but its width is equal to the span. In this case F2 = 0, so F1+F2 = -Lift, and so delta(M) = -Lift. The momentum change set up by the downwash of the vortices is indeed exactly equal to the lift. So the "lift=downwash momentum" people are correct in this sense.

So everyone is right some of the time, OK?