Hawk,
I think that the fault in your argument lies in the fact that the thrust does not remain unchanged as you climb. Thrust is proportional to mass airflow multiplied by the acceleration the engine gives to that air. The acceleration is equal to exhaust velocity minus TAS. As we climb the air density and hence mass airflow at any given engine RPM reduce, causing thrust to reduce.
In addition to this, the increasing TAS to CAS ratio causes the acceleration we give the air to reduce. This again reduces thrust. So even if we keep the same CAS and hence the same drag, excess thrust and climb gradient will decrease. This process continues until we hit the absolute ceiling at which maximum thrust equals drag, and excess thrust and best climb gradient are zero.
We can visualise this effect by drawing a drag and thrust available curve on a single chart. The drag curve will be the typical bucket shape. If we simplify the situation by assuming constant thrust at all airspeeds, thrust will be represented by a horizontal straight line. (In reality it would be a bucket a bit like the drag curve, but much shallower).
At low altitude the thrust line will be some distance above the bottom of the drag curve and will cross it at two points. These are the minimum and maximum speeds for which sufficient thrust is available. Excess thrust is the vertical distance between the two lines and this will be greatest at Vmd. Excess thrust is proportional to best climb gradient so this will occur at Vmd.
If we have marked our speed range in CAS we will find that the drag curve does not change with changing altitude. But as explained above, the thrust available will decrease as altitude increases. This can be represented by repeatedly drawing further thrust lines, each lower than its predecessor. When the thrust line just touches the bottom of the drag curve we are at our absolute ceiling.
It is tempting to think that we can treat excess power and excess thrust as if they were unrelated. This is not the case. As we climb, the power required increases and the power available decreases. This causes excess power and best ROC to fall to zero at the absolute ceiling. So at the absolute ceiling we have no excess thrust, no excess power, and the best climb gradient and best ROC are zero. Worse than this, there is only one speed at which we have enough thrust and power to fly. For a jet this is speed Vmd.
If we repeat the process described above, but this time with power available and power required, we can see how excess power and ROC change. As altitude increases the power required curve keeps being repeated ever higher. At the same time the power available line rotates clockwise about the origin, becoming shallower with each increase in altitude. When the power available line just touches the power required curve we are at the absolute ceiling. We have just enough power to fly straight and level at a single airspeed. If we superimpose the power and drag curves on the same chart we will see that the single speed at which we have enough thrust to fly is also the single speed at which we have enough power. For a jet this speed is Vmd. At this altitude it should really be called Vonly, but for some strange reason it is not.
Bookworm,
Regarding your comment "Surely the drag depends on the dynamic pressure and therefore the CAS squared?"
We need to be a bit careful here. If we change the CAS at any given altitude and temperature it is true that drag is (in part at least) proportional to CAS (or more correctly EAS) squared. But it is more complicated than that, because total drag is made up of induced drag and profile drag. Induced is proportrional to one over EAS squared and profile is proportional to EAS squared. Worse still the coefficient of induced is proportional to one over the fourth power of EAS and the coefficient of profile is more or less constant at most angles of attack. The overall relationship is therefore a bit complicated.
But the situation we were considering above was one of increasing temperature. In this case if we keep the altitude constant, we see TAS increases while CAS remains constant.
Looking at the drag equation Drag = Cd 1/2 Rho V squared S, the 1/2RhoVsquared is the dynamic pressure and the V squared is TAS squared. The ASI captures dynamic pressure and gives us a CAS indication proportional to it. So if we fly at constant CAS we are flying at constant dynamic pressure. But if temperature increases, the reducing air density Rho must be matched by a corresponding increase in TAS squared, in order to keep the dynamic pressure and CAS constant. This is the cause of the increasing ratio of TAS to CAS as temperature increases.
So as temperature increases at any given CAS, the drag stays constant but the TAS increases. Power required is equal to Drag times TAS. As temperature increases at any given CAS, the drag remains constant but is multpilied by an increased TAS. Because TAS appears as TAS squared in the drag, when this is multiplied by TAS to give power required we get something that is proportional to TAS cubed. The same thing happens as altitude increases.