Great piece Martin - especially the references to county games which, perhaps, say more about his credentials than his test career.
At test level, he was indeed lucky for all the obvious reasons - lots of weak opponents, a very fine attack, Botham in his pomp, Boycott back and never facing WI. Perhaps his first series when Aus were seen off 3-0 was actually his finest. I certainly think he was lucky in 1981, especially at Leeds, when Aus should never have made anywhere near 400 in their first innings.
Mind you, it was ripe of Illingworth to make anything of that, as his situation was similar - SA removed when he would have had to face them, relatively weak opponents for the first few years of his tenure, and then came unstuck when India & WI improved.
Obviously there's no definitive answer to the question in the title. My answer is usually Hutton, but I know he has his detractors. Before my time, of course, so I'm only going by the scorecards.