Great to see you here. I've talked about this many times with ND people of the era and, of course, read the great Twombly book, but I always thought the thing that rang most true was that, yes, Fr. Hesburgh and your father somewhat butted heads, most if it having to do with Fr. Ted establishing his vision of ND as a first rate university along with a first rate football program, but that your father's health, mental and physical, had simply became too worriesome.
As it was explained to me, Fr. Hesburgh did cut the number of scholarships available to your father, which certainly made things more difficult, but it was mostly because Fr. Ted simply didn't understand how hard it is to win in football. His actions were born more out of the desire to be well within the then current rules AND to win. It was as if he just thought that at ND you rolled out the football and the rest took care of itself.
But more than anything was that your father was so obsessive that people feared for his health. There is the example of fainting during halftime of the GTech game (they gave him the last rights), which they called pancreatis or something, but appears in retrospect to have been a massive panic attack. They didn't know that then, but that's what he was having. At least that's what a guy, a QB, on his teams told me.
It just became one of those things where he seemed to be getting more and more on the edge, more unstable in a way (moose Krause makes mention of this in the video Wake Up The Echoes), and when combined with the "fainting Irish" criticism and the other controversies that seemed to be coming up more and more, it just became something that "had to be done."
My source said, "It was also very typical of Father Hesburgh at the time that he thought a 25 year old with no head coaching experience could run the program. Father Hesburgh seemed to think the football program almost ran itself."
And so it would appear that, yes, your father was fired. Yes, there was some head butting (especially after) but that the main reason was the fear that staying on the job would kill him - or that he'd lose it.