To answer your questions, no the lifting forces of the keel or centerboard does not stay constant, it slowly drops off as you bear away from the wind. Because of this you can't really say there is a point at which the lift due to the keel is no longer useful to VMG but it does taper off to nothing at DDW.

Bot the sails and the keel work like an aircraft wing and they have both lift and drag. The wing works by creating a low pressure on the longer side, that is the side the air (or water) has to travel further to get around. If you imagine two particles of air that are side by side at the leading edge and are split appart by the wing, one traveling along the longer side and one traveling the shorter side then they meet up again at the trailing edge. Now think about a whole row of particles on either edge, the ones on the longer side need to speed up and spread out on the longer side than the ones on the shorter side. This creates a low pressure on the longer side which generates lift. due to the angle of attack the pressure also generates drag, along with skin friction drag.

Consider these forces acting on a sail. Close hauled the lift pulls the boat forward and the drag pushes it sideways and contributes to heel. The closer you sail to the wind lifting force also tries to push the boat sideways. As you fall off the wind and reach the lift vector starts to line up with the boat's heading reducing leeway.

The keel works the same but opposite, counteracting the sideways forces of the sails. As you mentioned, a flat board can generate lift as well as long as there is some angle of attack. However a flat board will stall out much quicker than a well shaped foil. When this happens your lift drops and you'll get more leeway.