If you look at the last decade, the first 5 years Arsenal had a team that was more than capable of challenging for trophies, it's only been recent years when they've lost key players, so that's not really valid reasoning.
Money and players are not solely responsible for winning things as Man City and Chelsea have found out this season, the manager plays an important role and as Chelsea have proven in the past regular managerial changes can be effective, something which may work to Man City's benefit next season.
Managers might play a small part in it but they're never going to win a league on their own. Martinez is a good manager but Wigan won't win the league any time soon. If he was manager of Man U for 3 years though I'd expect him to win the league at some point.
Arsenal obviously are a lot bigger than Wigan, but they're also some way smaller than the Manchester clubs and Chelsea who are the only three clubs who can realistically win a title in the next few years without something changing drastically.
Arsenal not winning the league isn't down to their manager, it's for the same reason as the other 89 clubs that haven't won it recently, they aren't good enough. Wenger, tit as he may be, isn't a bad manager just because he hasn't won a trophy recently.
So going back to the point, managers should not be expected to win a trophy every season because there's a lot of competition and only so many trophies. Far too many clubs get too big for their boots and throw a strop when they go a season without a trophy and sack their manager, forgetting that they've actually been doing well for their size or sometimes even over-achieving, and then they inevitably suffer from the disruption that sacking a manager causes.
Thankfully, it's that very principle that will keep Leeds out of the top flight for the next few years. They aren't good enough for promotion yet, and every time a manager fails to get them promoted he gets sacked and they try someone else, starting the whole project again.