You need both kinds of instruction. As to who was the better pilot, I'd say "We haven't seen either of them fly." If your father was in the air force flying a plane in war conditions, then he would have also been trained, and one way or another had this checklist in the back of his head. He transmitted to you this oneness with the instrument (plane in this case) and changing surroundings so that you could fly focused on your goal without distraction. Your mother tried to give you the knowledge you needed to draw on but I might question how it was done. To pull out a checklist in the middle of a flight and "teach" it then doesn't make sense since we cannot retain it all. I suppose if it happened flight after flight, you would retain more of it each time until it would be there for you subconsciously. But I'd rather be studying the items of the checklist on ground first: I'm not a pilot so I don't know how it's usually done and put into practice.
The analogy to piano is clear of course. I'd say that as a student I want both the naturalness without fear, and the knowledge. A teacher who can balance out these two elements while taking into account the nature of his student and the stage he is at is probably a gem of a teacher. And of course, we're assuming that the student actually wants to learn to fly.
OT to the actual theme: you might like to read the account of James Heriot, the veterinarian who was called up during WWII and had flight training. He recounts how his flight instructor made him a nervous wreck by barking out instructions and berating him for errors which often were caused by nervousness at the barking out. Then one day he told Heriot that he should fly out on his own, and when he did, he got lost over some field, found his way back and had to land under tricky conditions. All the stuff his instructor had barked out at him came to his mind and he landed safely. The same instructor who had told him what a dreadful pilot he was, was sitting in the canteen calmly sipping coffee. He said something like "Good landing. I knew you'd be a good pilot."
I think that is one way that music used to be taught, but apparently it created casualties as well as successes. How many students quit piano either for decades or forever?