Also, that perception of Fighter being some stupid, brainless schlub who can barely hold his weapon needs to fucking die. That is NOT a professional Fighter. A Fighter is a freshly Knighted Squire, trained with weapons and other skills. Or a mercenary soldier, just off his first tour, realizing that he has both the talent, drive and need to keep fighting, also well trained and moderately skilled, in not just weaponry, but the lay of the land and maybe even the political situation (A Fighter can probably be as well traveled as the Bard class, from day one.) Or a talented farmboy/girl, but she/he's got one thing all the other kids don't, courage and a willingness to fight. Or the man/woman who lost everything, and is out for vengeance, and again, is either naturally skilled, talented or even properly trained. Or any number of such backgrounds, but not the common, scared militiaman.
The useless schlub you're apparently so enamoured with is not a Fighter. He's the poor sap that the cowardly Lord of the Manor sends to die, while his Honour waits for the King's Knights to come and save his fat arse from the goblin horde. And the poor schlub is most likely going to die. But if he survives, he's not going to turn into an adventurer, he's going to run home, go back to the farm, go back to work, and hope to all that's holy that he never gets called to lift a spear ever again. Until the next time. And if he keeps surviving, he's still never going to be a Fighter.
But the fighter at first level
is a raw recruit, a greenhorn, very mortal and weak. Of course, some of how he ends up depends on ability scores--a fighter with an 18 Strength and 16 Constitution is more likely to survive to become a legend than one with 14 Strength and 11 Con.
The fighter at third level is a journeyman. He has survived "seeing the elephant," and has some skill developed. If he comes from a farming village, he is probably a man of some (local) renown. He is still mortal, of course, and far from a master. He perhaps has achieved the level of squire, or the cultural equivalent.
The fighter at fifth level is developing both a reputation and considerable skill at arms. He is a proto-legend, a man of renown beyond his village.
The fighter at seventh level...well, now we're getting a lot closer to your grand vision of a warrior. He has probably been knighted (or given the cultural equivalent of knighthood). He is a man of regional renown, and his name is spoken in the courts of barons and counts, perhaps even dukes.
The fighter at tenth level is your definition of a warrior hero. His puissance at arms is legendary. He is a master, on his way to become a grand master. Certainly he has been knighted, and quite probably given his own fief and post in the monarchy. Indeed, he probably spends almost as much time with matters of administering his growing domain as he does in the field. He is a "go-to" man for the nobility, even the Duke himself, and he is now being noticed and spoken of, if in passing, in the halls of the King.
The fighter at 15th level is a grand master. Now we are talking an Achilles, an Alexander, a legend known far and wide across the land, who can raise an army to command for his King. Or...perhaps...he could lead an army to seize the throne for himself, were he so inclined.
Obviously, the structure here is pyramidical. You have a lot of wanna-be lords and masters. Most won't survive. Some will be injured, maimed permanently, and retire from adventure. For every 1,000 1st-level chubs, there are maybe 100 3rd-level. 25 fifth-level. 10 seventh-level. 3 to 5 tenth level. Maybe 1 15th-level. The rest...their bones are strewn across the landscape. They didn't make the grade.
Obviously, you don't want that level of mortality in a D&D campaign. For the purposes of crafting heroic fiction, we're giving our PCs a substantially higher chance of survival. But what I described is how a gritty fantasy world operates. Fighter can be heroes...but as AC/DC sang, it's a long way to the top if you wanna rock and roll.