For Pete's sake - he was asked a question and he answered it. Gay Byrne must have known the kind of answer he would get, because Stephen Fry has never posed as an obedient unquestioning follower.
You say his career would be nowhere without his problems. He was damn near nowhere with them. The conflict he faced (personal and social) led him into a life of juvenile crime, followed by depression and considering suicide. Yes, he has used his personal life, but so as to demonstrate to many young people that others face the same things and can rise above them.
He has a most intelligent and enquiring mind, and he uses that to entertain. If we had known nothing whatsoever about his sexuality (which he denied anyway at the start of his career) he would still be a genial show host and those who actually know the man, not just the role, would still say that he is a good man and a good friend.
And he has never used what is the most personal part of his life so far - that he has just married his partner, and hopes to live happily ever after.