The URL of the wave is a great illustration of a water particle in a deep-water wave.
The only difference in shallow water is the water molecules below the level of the trough are eliptical. However, the water molecules from the trough to the crest still move in a circular motion. You can see this illustrated here:
http://www.seafriends.org.nz/oceano/ocean04.gif
There is no forward motion other than the forward part of the circle. The only time the water molecules move forward with any significance is when the wave breaks, and of course the white water. But since it’s possible to surf on the part of a wave before it breaks, and we don’t actually surf on the part that breaks forward (except for floaters), I find it irrelevant.
The part I don’t think you are realizing, is the water molecules don’t have any significant power. A water molecule on the surface of the wave is no different than a surfer or a piece of driftwood. Look at the animation again. Pretend you are the red water molecule. You would move in exactly the same way. And you wouldn’t really go anywhere, just like the water molecule doesn’t. I still think you are think that the water is operating like a Flowrider, but it’s clearly not.
Lets consider an area of it (the moving water) of approximately 2 foot (.6 meter) by 3 foot (.9 meter), or 6 foot square (.54 meter-squared) not exactly the area of the rear half of a shortboard, but in the ball park. A pressure of 1000 Newton per meter-squared impacting on an area of .5 meter-squared (pressure times area gives force) could potentially produce something on the order of 500 Newton, or about the same amount of force required to lift something or someone that weighs about 112 pounds.
But it’s not the water that’s doing the lifting, it’s the wave. The water is what’s being lifted.
That's a lot of pressure for water, in which the particles are moving at speed of 1 meter per second.
Surfing can be a gravity sport as in a sleigh ride, downhill ski, or snowboard, but most of the time propulsion is derived from the interaction of flow and the bottom of the board. The flows to do it are there.
It’s irrelevant, because that’s not what’s happening, but it’s not even close to enough speed. Take a look at the Flowrider, which is a good simulation of what you are talking about. The flowrider requires the water move at a speed of 30 mph versus the 2 to 3 mph you are talking about.
In fact the velocity of the water particles vary over the whole cycle of the wave form, of course the concern in surfing is only with the face, or leading part of the wave form.
Are you suggesting that the water molecules are moving faster on the front of the wave than the back?