Duncan Fletcher quite clearly disproved that though IMO. Yeah, sure, you can overestimate the value of a coach - no-one can turn crap players into good ones - but a coach is very important as he can turn potentially good players into good ones, who otherwise might have gone to waste.
Equally, he can gel things together - a point John Buchanan made well. IMO Buchanan holds-up well compared to Chappell. If "in times of change, learners inherit the Earth, while the learned find themselves beautifully equipped to deal with the World that no longer exists", Buchanan is the learner, Chappell the learned (at least on the subject of cricket coaching). Though it served well for Illy too.
The coach is vital as the "head of the family" as Buchanan put it. This is a role Duncan Fletcher played beautifully, and hence his team in 2000 and 2000\01, with most of the same players who'd been playing for the previous decade under Illy, Bumble and Keith Fletcher, achieved something none of the previous teams had looked like doing. Of course, DF was useful in other ways too, even if, like most, he had his foibles (tec constantly goes on about his over-insistence on the Sweep and one or two other things; I was frustrated by his inability to break free from the perennial "your best Test players are your best ODI players" nonsense which most of his generation and the next tend to believe; and there's no doubt he went too far occasionally on things that, for the most part, were mostly good ideas).
A crap group of players is a crap group of players. But that's emphatically not what England had in the 1990s, there were many potentially very fine players around. For any number of reasons, no-one had ever managed to quite harness that. Now, sure, he probably had some good fortune - there must have been an element of it-had-to-fall-into-place-eventually - but it can't entirely be coincidence.
A good coach (and general good management) cannot change poor players into good ones. But a poor coach (and general poor management) can change potentially good players into lesser ones. And that's what was happening pretty much between 1992 (the end of Micky Stewart's tenure) and 1999 (the start of DF's).