I'm going to ignore your section on aircraft and attempt to answer the (vague) question,
Do inherently unstable systems desire to be stable for all cases when a closed loop control is implemented on them?
First, I'll say that system response, stability, etc., are all based on mathematics, and math does not have feelings. That is, a system doesn't "desire" anything - it just is.
Second, there is a difference between controllability and stabilizability. Just because you can't force a characteristic to be what you want doesn't necessarily mean that it's unstable.
Similarly, just because you have feedback, that doesn't mean that your controller is stable.
Consider a bang-bang controller for steering in a car. You get feedback of the car's position in the lane, and you either steer full left or full right depending on which side of center you are.
A bang-bang controller is a closed-loop controller. The stability of the car's position depends on sampling and actuation frequency with respect to the car's speed. After a particular speed the position of the car no longer converges on the center of the road.
Another example would be a system where I want to control speed and altitude of a car, and I have speed and altitude measurements, but my only method of actuation is cruise control. Cruise control can force speed to go to whatever value I want, but it can't influence altitude. Therefore, I rely on the system describing altitude (briefly, $F=ma$) to be stable and, as long as the car has sufficient mass and the speed is relatively slow, that is true - the car's altitude won't randomly fluctuate. That is, speed is controllable and altitude is stable, so as a whole, the speed-altitude system is said to be stabilizable.
So, to summarize, a controllable system is one where you are able to force every state to the value you desire. A stabilizable system is one where you can force all of the unstable states to do what you want, while the rest of the states are inherently stable.
Output feedback doesn't guarantee either controllability or stabilizability. They instead depend on how your actuators are able to uniquely influence the states you care about.