To start, if you are sailing directly downwind, the faster you go, the lower the apparant wind on your sails. You are out running the wind. So far, so good? The faster you go, the less you have to work with heading downwind.
This is correct from a boat point of view. In reality of course, energy from the wind is being transferred to the boat and then bled off in hull drag. But, yes, from the sailor's point of view, the wind dies off as you accelerate downwind. There is less push on the sails and rigging. Naturally though, the rudder will still have good authority because you are moving. It's interesting that this is why a jibe is such a trap. You are heading downwind and the apparent wind seems very light, not any cause for worry. But if you veer off then the apparent wind speed picks up and hits your sails with unexpected force. You suddenly have your hands full.
If you sail upwind, you actually create more wind to work with (increased apparent wind), because you compound the wind as you begin to move into it.
Well, again, from a boat centered point of view this is correct. The increased wind will push harder on the sails and rigging. There will appear to be more energy available to drive the boat forward.
Therefore, it stands to reason that you will get more boat speed sailing into the wind, then away from it.
Yes, indeed, IF (and this is a big if) the apparent wind angle stayed the same.
So why a difference in close hauled and a beam reach? A beam reach remains the lowest point of sail, where you are still being pulled by the lift of the sail, rather than being pushed from behind. Therefore, you may still be creating some increase in apparent wind.
Now the vector part. If you thnk about where the boom is on a close haul vs. a beam reach, this may begin to make sense. Since you are creating a wing with your sail, the boom is much closer to the center of the boat on a close haul, as the wind is coming more directly at our bow. On a beam reach, your boom is eased out so the front of the wing (sail) is facing the wind coming from the side.
I think what you are saying is something like the following. If the boat were stationary and facing perpendicular to the wind then the wind would be coming directly over the side of the boat. And, in this position, 100% of lift from the sail would be used to push the boat forward. Here we have maximum potential (but of course the boat isn't moving yet).
As the boat begins to accelerate forward, the apparent wind will increase and this would theoretically give additional lift to the sail. And, it would if the wind angle stayed the same. However, it doesn't. As the boat accelerates the apparent wind will come more and more from the bow. It will shift forward. And, the apparent wind will shift more forward the faster the boat goes. This is bad because, as the wind shifts forward, it becomes less useful. Why? Because lift is perpendicular to the wind and therefore as it shifts forward an increasing amount of the apparent wind would only generate side force rather than forward propulsion. Also, the increasing side force will have to be countered by the keel and rudder so this will increase drag.
So, why is close hauled worse? As you pull towards the wind the apparent wind vector shifts even more towards the bow, so even though theoretically the apparent wind would increase, there is less and less lift to move the boat forward. More and more of the wind's energy simply goes to pushing the boat sideways which would result in more heel and more leeway but not more forward motion.
I think this is what you were saying and I agree. However, even though I tried to say it as simply as possible I still ended up with three fairly complex paragraphs. And without benefit of diagrams this can still be misinterpreted. I think this thread has included a lot of saying the same thing in different ways.