This is amusing, in the not too distant past such heavyweights as Alan Pardew managed West Ham and Crystal Palace, Tony Pulis managed Stoke City, Tim Sherwood managed Aston Villa, Steve Bruce managed Hull City, Avram Grant managed Portsmouth (who finished bottom of the Premier League), and Championship sides Millwall and Cardiff have all reached FA Cup finals.
Although none of those sides took the final step and lifted the cup, the fact that so many teams in the last 15 years or so - whose odds of winning the competition were as wide (or even wider) than Wigans' - have reached the final proves that winning it is not like Kenya winning the world Cup - as you claim. Not even close.
Alan Pardew's two sides for example took the lead in both games, he could quite easily be sat at home with 2 FA Cup winners medals and absolutely nobody would have a different opinion on whether he was a good manager. Perhaps he would have won both finals had he been playing against a 10 - man Manchester City whose squad knew that the manager was getting canned regardless of the result...
Martinez, across 3 seasons, was not a good Everton manager. The positives he achieved in his first season are easily undone by the miserable lows of the following two. No Evertonian wanted Martinez gone because of anything Moshiri had done, we wanted him gone because he had failed. End of story.