Yeah, that's pretty sick!
Self-dampening of the player's SFX in COCKPIT mode does sound useful and was requested before. I'll try to look into it once I get some other stuff sorted out. Hopefully I can find some way to get it to work with being able to toggle between modes.
You kinda hit the nail on the head with the last part. Stability was intended to reward players who stayed on the ground more. They could fire faster and more accurately and could absorb hits without flinching, especially since you recovered stability very quickly on the ground. Some of the heavier weapons even rely on being grounded to be effective. On the other hand, I felt like aerial players could already take advantage of their speed for hit-and-run tactics to not be affected as much by low stability (shorter engagement times, closer ranges, etc).
On watching your stuff, you didn't seem to be having any particular issues with stability, even during the segments where you were chaining jumps, bursts and kicks. It seemed more to happen when you were under sustained attack, like against Boulder during the second stage, or when you lost all your ARMOR after that engagement (ARMOR also reduces stability damage as long as its non-zero when you take damage). At these points, it's usually wise to try and gain back some stability in cover rather than run it red and lose out on offensive capability.
And yeah, the Large Missiles basically live up to their legacy in the AC games by being broken af. xD