Keith, good explanation, but like I said, in practice there is a difference between theory and practice.
For a simple fixed pitch prop and a normally aspirated piston-prop aircraft thrust is maximum before the start of the take-off run, when the air is given the maximum acceleration.
I don't think this is necessarily true. At zero IAS part of the (fixed pitch) prop may be stalled as the blade angle (particularly near the root) is above the stall angle. Only once the aircraft starts moving and the IAS becomes larger than zero will this part be unstalled, which actually increases the thrust.
At the other end of the spectrum, at very high IAS, the tip of the blades may actually not be able to keep up with the airspeed, and may even generate a backwards force. Although my gut feeling is that fixed pitch propeller blades are designed so that this would only happen in a dive, at speeds well above normal cruise speeds.
So even at fixed RPMs, propeller thrust is far from a constant. In fact, there will probably be some sharp corners in the propeller thrust curve, which happen at the points where parts of the blade become unstalled, or go into beta.
As the aircraft accelerates down the runway the difference between prop wash speed and TAS decreases, so the thrust decreases. So the thrust curve starts at a maximum when TAS is zero, then curves downwards as TAS increases.
Another factor is that as the IAS goes up, the total thrust goes down, like you said. But that decreasing propeller thrust goes hand in hand with decreasing propeller drag. Which, in turn, means that the engine will turn faster. Which means that more air and fuel is sucked in and burned. Which will mean that the power exerted by the engine goes up. Until you reach an RPM where the combustion process is not finished when the exhaust port opens, and engine power actually goes down.
This all means that you now need to consider the engine power curve itself too, which alters the equation. I'm not arguing that the curve doesn't curve downwards or anything. I'm convinced the basic shape stays the same. It's just that the exact shape of the thrust curve is not just dependent on TAS and prop wash, but also on propeller efficiency and engine power output at various RPMs.
Adding a turbocharger will make things even more complex because it complicates the engine power curve to a great extent. While adding a constant speed prop may simplify things since you can now take the engine power curve out of the equation. And so far nobody has mentioned the mixture, which will alter the engine power output independently of RPM, IAS, density altitude and whatnot. (And you cannot simplify the model by assuming an optimum mixture - whatever that is - across the whole curve, since at certain power settings, an optimum mixture will cook your engine.)
It's stuff like this that will screw up the nice theoretical model.