Parmandur
Book-Friend, he/him
I agree to an extent. The fighter can satisfy some of that fantasy. As you reiterate, they can take and make hits well. Where I, and perhaps others in this thread, take issue lies in the how.
First, being able to take hits when that is expressed almost entirely in HP or AC is mostly passive. In contrast, for example, a wizard can throw up a Shield spell to adjust their AC. This does not strike me as particularly interesting in terms of imaginging a high level class fantasy of the fighter. Second Wind is probably one of the more active abilities for fighters in this regard. So here I would invite you to consider with me how we could express "unkillable" in more active ways other than HP and AC, particularly for a higher level class fantasy for non-spellcasters.
Second, "being really good at shooting things with a bow" often entails a lot more than shooting quickly and accurately, but also a variety of trick shots (or even trick arrows) that empowers these sharpshooters to have more narrative control than they would otherwise. Their abilities are also often accompanied by other associated feats of physical prowess (e.g., freakish athleticism, "hawkeyed" sight, etc.). It's also hardly a coincidence that a lot of the mundane or non-powered heroes are often ones that have other accompanying narrative benefits (e.g., the power of being SUPER RICH). But one basic fantasy for these archer archetypes is the ricochet shot. Can a 5e fighter archer perform a ricochet shot around corners? Can a 5e archer aim or make a called shot that expresses their capacity to hit with their arrows on a more localized location? To the best of my knowledge, no. The best bet would be to create a subclass or a BM maneuver capable of doing it.
This is likely why the 5e Battlemaster is regarded as the best fighter subclass. The maneuvers provide the greatest amount of narrative control options, at least in combat, for many mundane classes. But it is one subclass for one class. But this is baked into the BM subclass as opposed to the proper Fighter class. And whereas magic users typically receive more new spell options in new books, the BM maneuvers don't. A trend that is mostly par for the course when it comes to martial options. (New feats and subclasses are something equally available for magical classes.) And uncoincidentally, the 5e battlemaster is probably one of the most 4e like archetypes in the 5e game.
I prefer playing a Champion, personally. Nice that the is an option, though.