The Fighting-Man is the default protagonist, but there are other archetypes. They arose out of defining themselves against the Fighting-Man, often in the form of a tradeoff: he's really good at (X) but terrible at (Y) because (Reasons). The first of these to arise was the man who succeeds in his challenges through his superior wisdom or cunning- he out-thinks his opposition. I'm making this simple and call it "The Thinking Man".
He's defined by not being a Man of Action. He's not the sort to throw down, and his chase performance pretty much relies on him not being on foot. He relies on something that represents his superior intellect, wisdom, or cunning to handle any situation that would get a Fighting-Man into displays of athletics or skill at arms. Here is your Wizard, your Shaman, your Gadgeteer- and your Trickster.
You'll find this character archetype as the protagonist in genres of fiction where being a man of action isn't a strength as such, usually mysteries (e.g. Detective stories) or horror, and in other genres you'll routinely find this archetype in a supporting role for a Fighting-Man protagonist.
In more recent years, there's been blurring of the lines, but this is the sort of reveal a story with a Thinking Man protagonist builds towards- and as such it's the sort of thing the audience expects when you feature such a protagonist in your story. Do not disappoint; the strength of the reveal will make or break your story. As seen here in this classic Charlie Chan film at 1:00:10, you get the climax where the Thinking Man protagonist brings it all together- at which point you will see the foreshadowed elements in retrospect.
Be it a mystery, or some other puzzle--and there should be some form of puzzle--requiring his intellect or cunning to solve, there is always a moral element to the matter. The power of the Thinking Man is not only knowledge of expert topics, however narrow, but also observant of people and patient in making use of those observations. Gandalf the Grey displays this in his experiences, where his use of power is tightly constrained and so much rely on lore and people skills more than his inherent power (or that of the Ring he's entrusted to use) to solve matters.
Competing Thinking Men as Protagonist and Antagonist, properly executed, produce brilliant storytelling via their Duel of Wits- each trying to outsmart the other in order to adapt and overcome the opposition. In popular media, you won't find a better example than Yang Wen-Li and Reinhard von Lohengramm from Legend of the Galactic Heroes:
This is different in its execution, but it also allows contrasting personae to facilitate both plot advancement as well as character definition and development. Yang is cerebral, but casual and modest to a fault; Reinhard is imperious, impatient, and far more ruthless than Yang. Yang's the better reader of character of the two, and it allows him to upset Reinhard time and again- despite him being infamous for being useless below the neck. (A true non-action guy he is, whereas Reinhard has some chops as a Fighting-Man. Some.)
You will also see this with ensembles as the leaders of competing factions--Charles Xavier vs. Magneto, for example--but these are usually positioned, in narrative terms, as supporting roles (what TV Tropes will call "Big Good" and "Big Bad" respectively). Their Duel of Wits fades into the background but their moves form the basis of the actual protagonist's stories, creating opportunity for these masterminds to become protagonist/antagonist at a meta-level; challenging, rarely pulled off, but when it does you get genre-defining examples such as Mentor of Arisia vs. The Innermost Circle of Eddore (to whom Gharlane of Eddore played Starscream to its Megatron).
It's not as broadly applicable as the Fighting-Man, but it's pretty broad and with the right support and execution you can do a lot with it. You'll find this a lot in Miyazaki films, where the (often female) protagonist has to rely on wits and charm to win the day, so there's your example of Little Thinking Boys and Girls. (Most famous being the Potter kids.) Done right, you get some of the best Doctors in Doctor Who; done wrong, you get the worst. Just remember this: the nature of how the protagonist deals with the conflict dictates the archetype you employs. (Never send Encyclopedia Brown to do The Shadow's job.)