Game day coaching is too big a factor in how good of a coach one is, and we simply didn't see enough out of Pagano last year to really know how good he is in that regard.
But I agree with Donnybrook about Mike Smith. The guy has a fantastic record as the Falcons coach, but I swear, every Falcons game I watch, he makes several inexplicably dumb coaching moves (many of which often do not cost his team, but they are still bad moves).
I would argue that game day coaching (with the very notable exception of play calling, which is frequently handled by an assistant, anyway) is by far the smallest factor in determining how good of a coach one is, it just happens to be the most visible.
What would you consider the biggest factor if not making sound coaching decisions on game day? Assuming that position coaches are doing the teaching.
If we assume that the position coaches are doing all of the teaching, then I'd say assembling a good staff is the biggest factor.
Ultimately, I view the head coach as more of a CEO than a coach. He's ultimately responsible for making sure his team is meeting goals in certain areas- player development, motivation, use of resources, scheming and installing a gameplan, playcalling, in-game management. Some coaches are more autocratic, while others are more delegators. Mike Shanahan is very hands-on when it comes to installing an offense. John Fox is not- he'll hire a coordinator and let that coordinator do his thing. Shanahan wants to run a very particular offense. Fox is very flexible (as the last two seasons have demonstrated). Neither answer is "right" or "wrong", the only thing that matters is the quality of the results. If a coach hires fantastic position coaches who teach all the fundamentals and fantastic coordinators who create amazing schemes, and then the coach goes and plays a round of golf, then that's still a wonderful coach. If the coach hires a bunch of unpaid interns and does all of the coaching and scheming himself, but produces similar results, then that's still a wonderful coach, too.
As I see it, the coach's main responsibilities are overseeing player development (i.e. making sure every player on the team is improving), roster development (creating a 53-man roster, deciding inactives, creating depth charts, fostering competition, and playing his best players), creating an overall scheme or "identity" (making sure every player has clearly delineated responsibilities and helping that player succeed at those responsibilities), installing weekly game-plans (creating wrinkles in the general identity designed to take advantage of the specific weekly opponent), executing those game-plans (calling plays), motivating his players (ensuring that players are executing as close to peak level as consistently as possible), and managing the flow of the game (calling timeouts, working clock stoppages, using challenges). I might be overlooking some things, but that seems like a reasonable broad-strokes job description. Whether he handles those areas directly or delegates authority, he is responsible for ensuring all of that gets taken care of.
Now, if we envision a coach who is absolutely fantastic at all of those areas except he's the worst in the league at player development, I don't think he'd be a very good coach- his team would just hemorrhage talent until it's the worst squad in the league. Likewise, a coach who can't manage the depth chart and play his best players is in trouble. A coach who doesn't create a team identity will never get anywhere, and a coach who is so vanilla and predictable that he doesn't tailor his scheme to his opponent will preside over a team that fades hard at the end of every season. A coach who is worst in the league at motivating his players will likely be ousted by mutiny before long. Meanwhile, if you had a coach who was excellent at every other area, but absolutely abysmal at managing game-flow, how good could he be? Well, Andy Reid managed to make 5 NFCCG appearances and one SB appearance in 12 years despite being pretty universally regarded as one of the worst in-game managers in the league today. John Fox went 13-3 last year and earn the #1 seed. Marty Schottenheimer and Bill Cowher both had amazing careers despite being "too conservative". A lot of these guys tended to underachieve in the playoffs (although Reid and Dungy combined to go 19-19 in the postseason, which is actually a pretty darn good record), but it's possible a lot of that was simply because the team overachieved so much during the regular season. If you take a squad that had no business making the playoffs, earn a 5 or 6 seed, and get eliminated in wildcard weekend, is that a negative? If you take a team that should have been a 4 or 5 seed, put up a huge winning streak and capture home field advantage, but then get eliminated early, is that a disappointment?
Either way, If I were going to hire a coach who was fantastic in almost every area, but who had a major deficiency in one, I would gladly take the guy who sucked at in-game management every single time. I would prefer if he was better at in-game management (or if he'd just hire someone whose only job was managing games), but obviously the guys who are great at everything (Bill Belichick, John Harbaugh, Bill Walsh) are pretty hard to find. If I was going to take a flawed coach, I'd rather his flaws be in game management than player development or scheme, because I just think game management is by far the smallest part of a coach's job description, even if it's the most visible.