You could couch your request for feedback in this way:
"Please tell me where I went wrong so that I can correct it and help imporove my chances in succeeding in my chosen profession."
If they still don't answer I would suggest you wouldn't want to work for them anyway. However, it may be that your letter is shunted into the wacky HR department who don't give two tosses about your career. If you can land it on the HOT's desk you should, hopefully, receive a more sympathetic hearing. It's such a pity that some of those in a position to make or break someone's dream do not refelct on what it was like for them and try to be more helpful.
But then the most used phrase in aviation is "life's not always fair."