I'd take better to calling them an Un-Hero. Non-Hero implies that they didn't do anything heroic. A Non-Hero would be someone that interferes in minor ways for the forces of heroism, but doesn't really do anything big. A protagonist who does not really affect the outcome of the story in any way. Indiana Jones in every movie except Temple, the original Star Wars heroes in the new Star Wars canon, The Narrator in The Great Gatsby, Shepard in Mass Effect, (if you subscribe to the indoctrination theory) Zatoichi in some cases*. Those are Non-Heroes. Un-Heroes are heroes that suck at being heroes. Something that's too hateworthy to be a hero, but has too much screentime not to be. In that uncanny realm between hero and shithead foil to better characters, like Un-Dead lies in the uncanny realm between living and dead, like all the shitty beastman units are called Un-Gor.
Because this thread is also about silent protagonists now, I think that, personally, I can potentially connect with characters that don't talk almost better than characters that do. There's just a very careful art to it. Take Charlie Chaplin and Buster Keaton for example. They were the greatest comedians of their time, yet their films still had genuinely sad and touching moments. If you took out the dialogue cards, you'd very probably still be able to tell what's going on. Communicating without explicit words is just as valid a form of communication as any.
Take 2016's Doomguy for example. no other media in the world has illustrated such casual disdain, illustrated simultaneous competence and awareness as well as nonchalance, and at the same time created a goofy, witty personality while maintaining that same imposing and intimidating presence. Doomguy has all this without uttering a single word. If he had talked, it wouldn't matter how cleverly he was written, there's nothing that would make him half as endearing and badass as the stark quiet behind the animation.
Doomguy doesn't need to be social, he knows himself and doesn't care. You're perfectly free to engage with him, his personality is readily, vibrantly apparent in everything he does, but conversation is not something he needs or wants. Bethesda has filled out a character that literally needs no introduction, bringing personality back to a silent badass that spawned a long line of one-liner-spewing supersoldiers without making him anything other than the silent badass he always was.
I wouldn't have felt the same giddiness at fist-bumping an action figure, not even if this was a silent moment between Doomguy dialogue. It would have felt quaint, but not quite as intimate and relatable as silently exploring a person's attitude, opinions, and quirks. Outright talking creates a degree of separation. It makes the situation a social thing, rather than pure call-and-response establishment. Coyote and Roadrunner are some of the most complex and interesting characters this side of Batman, and they've only ever established themselves through actions and a few signposts.
I know it doesn't technically count because some do have vocalisation and dialogue in some cases, but take the Styg, the Silent Grandpa from Extremely Loud and Incredibly Close, Raymond Teller's stage persona, Ninja Brian, Groot, Fury Road Max, etc. etc. None of them are protagonists, but given a brief change of story, they easily could be. Hell, I guess Teller, Groot, Mad Max, and Ninja Brian sort of are protagonists, (Or, at least deuteragonists) and sole protagonism really wouldn't be that far off if the world didn't place such serious emphasis on the importance of dialogue. It REALLY isn't needed to create a deep and compelling character.
*Not to undermine the statement that he makes by killing everyone, but a lot of the time he comes between two factions of assholes to kill them, it's pretty apparent that they were either preparing to kill each other or were going to kill each other before Zatoichi interferes. There's quite a few films where Zatoichi does nothing but impart wisdom and kindness to strangers and old acquaintances, maybe save one or two lives, and then climactically walk into the middle of a gang war only to murder everyone by himself. The ending would've been much the same, but the fact that Zatoichi does the killing because he is the one to do it changes the tone significantly. Which is weird, in an era of Wuxia and Chanbara and Westerns, where everything's black and white Zatoichi as a non-hero sort of calls into question the moralss of those kinds of movies in general. If two forces are killing each other, is it right for one righteous dude to walk in and start killing both of them to protect the good guys in a world of gangsters? Do means and motivations matter when the ends are the same?... God, I love binge-watching old movies on youtube...