When those called to lead refuse God's call, people get stuck with bad leaders. For every bad leader, three people may have refused God's call to lead (see Jgs 9:9-13). For every buckthorn, an olive tree, fig tree, and vine may have refused to make the sacrifices necessary to answer God's call. When our leaders come from the "bottom of the barrel," it means that those from the top and the middle of the barrel are disobeying God by refusing His call.
I have seen this phenomenon up close. When the qualified and called turn away because they are too busy with other things, rest assured that there will be a mediocre and unsuitable person with a great desire to be the center of attention ready to become "the boss" of a particular ministry. And it is very hard to get rid of such mediocre people once they fulfill their dream of being in charge of something; their grip is very tight. So, when out of false humility, you seek to disqualify yourself from leadership--maybe, you should take time to think of the alternatives that will quickly and without hesitation step up to the plate and wonder which is worse: the mediocre alternative or you yourself, regardless of your known defects and imperfections. I seem to recall a similar insight in a play about St. Thomas More in which someone (I think the saint himself) took on a leadership role in part because of the looming alternative if he did not do the job himself.