I'll be honest--I don't specifically know what it is that I'm looking for, with the "more complex" Fighter type. Particularly with the relatively extensive list of caveats about what I'm allowed to reference. It's a little like Justice Potter's response to someone expecting him to precisely define "hardcore pornography": "I shall not today attempt further to define the kinds of material I understand to be embraced within that shorthand description, and perhaps I could never succeed in intelligibly doing so. But I know it when I see it, and the motion picture involved in this case is not that."
So: I don't necessarily believe I need to lay out precise figures or characteristics. But I can distinguish where it appears (which I have been requested not to speak of), and where it does not appear, namely the 5e Fighter. The Battlemaster has an idea of it, but it's not quite there, much like a film which happens to depict sex between two people has an idea of "hardcore pornography" but isn't quite there (and thus cannot be prohibited on merit of that alone.)
However, I think I can speak about what things help keep me engaged with a character when I'm playing it. I have no problem whatsoever being engaged in the story, so I will speak no further of it--that has never been a concern on any character I've played, ever. However, I frequently find that being thrown into a situation with "no tools"--that is, *zero* well-defined mechanical elements to hook into--that what results is frustration, sometimes severe enough to make me want to walk away from the table (metaphorically, since I've only played online). Perhaps it is a quirk of my thinking processes compared to most people; perhaps it is purely a game-style difference or force of habit; I don't know. But when I have tried to find improvisational or so-called "creative" solutions to things, even in games where that is encouraged (like Dungeon World), I have almost without fail encountered at least one of the following:
- Excessively high difficulty level, such that the task is essentially guaranteed to fail from the outset.
- The DM does not properly understand probability, specifically iterated probability, yet forces the situation to be "you must never fail in any of an indefinite string of rolls." This means that, while the task may be easily accomplished in the short term, it is guaranteed to fail in the long term.
- The potential benefits of the improvisation are highly circumscribed, while the penalties for failure are extensive, thereby making it a risk not worth the reward. (I admit, here, that I am almost surely more risk-averse than the average player, but even risk-takers I've known have slowly learned that "the house always wins.")
As a sort of corollary/combination of the above three things, there is a fourth thing: playing as a class without access to well-defined resources, I have almost never seen a situation where I am able to achieve more than the most meager effects of classes that do have well-defined resources, even at fantastically high level relative to those meager effects. Yet, at the same time, success often hinges upon being able to achieve large-scale, dramatic, or long-lasting effects.
I am 100% on board with the idea that a class called "Fighter" (or "Warrior" or "Knight" or whatever else) should be at its best when fighting--when using the skills of arms and armor--or when performing feats of physical prowess. But my experience has been that DMs are a skeptical, "conservative" lot when it comes to areas of personal prowess (physical or mental) and things one can do in combat. Such conservative skepticism is reserved almost exclusively for those characters who lack access to magic (I would normally say "classes," but magic is as much a function of subclass as class), while characters who do have access to magic are treated to near-zero skepticism about the applications of their abilities--even ones not actually indicated by the abilities in question. (And yes, I've seen this happen even with an otherwise-excellent DM in Dungeon World too, not just D&D, though I was fortunate enough to be the magic-user in that instance.)
There's also a second problem, not directly related to the first but which still matters for me. Simply put: I get bored in "mechanical" situations, such as combat, if I have very few distinct tools to employ or choices to make. And when I get bored, my mind wanders; I lose track of what's happening in the combat, which slows everything down for everyone; and I try to "fill up" that empty cognitive space with some other thing, which may or may not end up helping. My experience of more..."choice-and-option rich" gameplay (which I cannot explain further, per the OP's request) has shown this doesn't happen all the time, though--games that give me "enough" choice (which I know is purely subjective) don't have that problem. But, at the same time, I really don't actually care that much for either the fluff OR mechanics of most casting classes (ironically, I am probably most strongly *like* a Wizard IRL, but I find the Wizard archetype tedious in D&D). I further tend to like playing physically strong, contemplative characters, often those associated with some kind of martial tradition (Paladins especially, but also Fighters, Warlords/Captains, and Monks). So what am I to do, when my only option for "choice-rich" characters is spellcasting, which I find tedious, and when my only option for "martial tradition" characters without spellcasting is not at all sufficiently "choice-rich"?
I don't think it's at all weird to enjoy a breadth and subtlety of choices, while at the same time not liking spells (the specific mechanical implementation) or spell-casters (whether mechanically or thematically). Yet for someone with such preferences, the Battlemaster is the closest you get, and it feels like a token effort.
So that's what I find wrong with the situation, and why I'd like to see it changed--to include an option for me that I find interesting and engaging, without mucking around with things I find tedious or which have flavor I don't care for. But I'm no game designer, so I don't know what it would be to do that, or how one would go about it, any more than I know how to fix a collision-detection bug in a video game. Yet I know I want this change and would almost certainly recognize something that did it, just as I know I want collision detection to work in a video game and knowing that a particular action shows successful or failed collision.