I am wondering if a boat moves because the wind pushes or if the sail takes the role of a wing and moves the boat because of pressure? My curiosity comes from sail shapes and whether the shape, or square footage has the final word in driving force.
That's a very complicated question.
Generally speaking, down wind (abeam or farther aft) the sails are just expensive bags that catch wind, which pushes the boat along. Up wind, the sails create a low pressure area in front of themselves, which causes the higher pressure behind them to drive the boat.
This is a very generic, but accurate way of looking at things. Many boats have such efficient hulls, appendages and sails that they don't ever truly sail down wind, unless dead before the wind, which they never do, if they can help it. These types of boats are creating their own wind, so to speak, or at least what the sails work in, is artificially increased wind speed, developed by several elements of the design.
Shape plays a huge role in the dynamics of sail effect, but there are many other factors too. A well burdened hull, one that is overly built and poorly shaped can be made to fly if enough sail area can be managed. The "sand baggers", used at the turn of the 20th century, in some cases had triple the amount of sail area a similar sized craft may carry. This extra horse power made them spectacular speed machines, but they had to be expertly handled and often broke or capsized.
No one item, in the many possible variables, governing sail performance really is responsible for a giant leap in performance. There are things that can be done to increase performance, but they include a list of many modifications that will need be incorporated into the boat. Weight reduction (especially aloft), windage reduction (especially aloft), sail shape, sail cloth, sail attachment, effectiveness of rig and sail shape controls, etc. The list can be quite extensive and if you are fleet racing and trying everything you can to maintain a winning edge, no dollar is spend on cup holders, just faster winches and such.