Well, part of the problem is that main characters have to succeed at everything, which means that any flaws they might have pretty much by defenition have to be overcome, irrelevant in the first place, or rare enough that they only run into it for comedic/dramatic effect. This can be done better or worse, obviously, but at the end of the day you're still looking at something alarmingly close to either "Main character is a god" or "Main character is a cripple who relies totally on others to perform certain tasks."
Of course, in theory this ought to be easier to avert with team-heavy works, since the "main" character doesn't necessarily have to accomplish/defeat
everything by himself, but I can't think of any examples.
That said, I'd just avoid loser/average main characters to begin with, since the whole "but wait they're really awesome at least sometimes!" thing is either the whole point or quickly becomes it.
He was just an average shy boy with parental issues until his father wrote him to come to Tokyo.
But see, the 'average' in this is just pointing out that he's not a mutant or something (yet). He's definitely not quite average mentally, and his father is... not the sort of person average people have as a parent. Hell, not even "come to Tokyo" has quite the same connotations in that world. Even before finding out his greater significance, the premise is a lot more interesting than "This is Joe. He's average. Then stuff happens to him. Woo!"
Hoshi no Samidare, the protagonist had savere grand-father issues that were actually well integrated with the character.
That also pulled off the "shut-in nerd to athletic warrior" transformation fairly well, since he basically just started training a lot because
horrible golems are coming to kill him so maybe he should train for that.Still didn't like it much past the beginning, though.