Your handwaving about the conservation of energy is unconvincing. Are you honestly trying to tell me that the work done per unit time of a rocket increases with its velocity? The energy for that work has to come from somewhere. This implies that the chemical reactions and fuel use by the rocket increases with its velocity as well (and don’t try to bring up mass lost because if you increase your speed ten fold, you increase your kinetic energy a hundred fold while your mass might only drop by a small amount). Is this what you are trying to say?
I most certainly am.
If you had taken high-school physics, you would know that "work" is "force exerted over a distance".
If you push very hard on a very big rock, moving it not at all, you have done no work, no matter how tired you are at the end.
"Power" is the rate of doing work.
If you increase the rate of distance covered (higher speed), you increase the rate of work and hence the power.
Slightly altering the numbers of my previous example, a rocket with 550 pounds of thrust, tethered to remain stationary, does no work. That same rocket moving at one foot per second does 550 foot-pounds of work per second, for one horsepower. That rocket at 36000 feet per second (roughly escape velocity from Earth) has the equivalent of 36000 horsepower.
P.S. - If you don’t believe me, call down to the local high-school and ask to talk to the physics teacher.