I honestly have no clue since I'm convinced you can make anyone into the perfect protagonist. If I were to give an answer my guess would be consistency and decisions that make sense? Unless the point of the story is that they don't make sense, then I don't know lol?
For some reason people like to put limits on what a protag can or cannot do but I'm like "Nah son. Go all out."
That being said, there's such a thing as being too edgy which makes this situation a slippery slope. (I'm looking at you Snyder/Family Guy)
But yeah, that's my answer. When I write, I like to make characters really likeable but they have traits/or have made decisions that make people go "That's not... hmmmmmmm".
Sometimes these characters change. Sometimes they change some of the negative traits, only to have other traits left over before they get killed or before the story ends. Sometimes they don't change at all. It's nice to keep audiences on their toes because in real life, you can't expect people to change so easily. Da deeds make the man. :v
Jojo is really good at this btw. James Gunn is also really fantastic with this type of stuff like The Suicide Squad (the good one) or Peacemaker lol. One of my favorite scenes in The Suicide Squad is when Bloodsport is surrounded by brainwashed innocent people and he's trying his hardest not to hurt them, but the last five seconds before he's saved he just starts nervously blasting because he was under pressure. Usually people just show the protag getting saved before they can really react to that point.
Btw if you want to cater to me specifically, scenerios that show the flawed side of the characters work (or if your character is perfect, scenerios where they just manage to make big brain decisions).