SPOILERS for Mana Khemia, Okami, Professor Layton and the Unwound Future, The Legend of Zelda: Skyward Sword
So this made me think of the end game of Mana Khemia and how Vayne breaks down at the end. He wanted to disappear because of killing his father and how dangerous he could be but he also didn't want to be alone and so he ended up making a wish to have him and his friends disappear together.
Okami had some badass guys but it also had its weak ones all over the place. Even Issun, your companion is tiny and not nearly as capable as he'd like to be. Susanoo is the perfect example of a coward but he eventually becomes a hero.
Professor Layton and the Unwound Future has the the two antagonists show weakness and sadness and Layton himself is clearly pained by losing the one he loves a second time.
Shattered Memories... Holy crap I wouldn't even know where to start.
Ghost Trick has a bunch of guys that are weak and/or cowardly, one of them is even a high ranking official.
Elebits: The Adventures of Kai and Zero, has you playing as a young boy and he gets emotional when certain things happen in the story.
Luigi's Mansion has you play as Luigi who is clearly scared out of his wits but he continues on in spite of being afraid.
In Skyward Sword, Link shows that he has insecurities when he fails to protect Zelda.
Rune Factory 3 has you play as Micah, and he's not really a manly badass, but he still manages to fight monsters, look after a farm, and find love without having to be.
The Megaman Battle Network series often has Lan hit moments where he's down and doesn't know what to do, often times bad guys will even trick him into doing stuff for them.
Megaman Starforce certainly has this element with Geo starting off as a socially withdrawn boy who avoids interacting with people so he won't get hurt by losing them like his father.
Rayman 2: The Great Escape has Rayman captured right at the start and at other times in the game he's shown to not be able to keep fighting if not for the help of his friends.
Tales of Symphonia has a bunch of male characters that don't fit into the archtype of badass (Though they do exist in the story). Lloyd actually breaks down a few times, Genis has a lot of baggage too, as does Regal.
I'm honestly surprised at how easy it was for me to list off examples.