Leadership is a feat. Fighters get extra feats, making Leadership a more attractive option for them.
Furthermore, there were several fighter-specific feats that rewarded a "tactical" character with a high INT. A whole chain of 'em.
Admittedly, the support was not very robust (much like the 2e support for thieves being assassins), but it wasn't absent. 4e supported the character concept more robustly, but it isn't clear that this mechanical support necessarily mandates a class. Change half of the 4e warlord powers to Expertise dice options (which are a lot like 3e feats), slap it on something like the Veteran fighting style, expand the ability adjustment of fighters to also include Intelligence, Wisdom, and Cha (which is actually kind of awesome anyway) and what more do you need?
Leadership scales off Diplomacy. Not a core skill for Fighters. I suppose they could use another feat to gain access to Diplomacy, but I think you're aware of how disingenuous your excuse is (nevermind Leadership being the most banned feat in the game, beating out stellar additions like Greenbound Summoning and Divine Metamagic). All the Int skills rewarded the fighter PERSONALLY with extra maneuvers, etc. Didn't do much for the fighter's party members.
The fact that your suggestion for Next to make the Warlord part of the fighter expanded the ability adjustment of fighters to 6/6 stats, completely changed their core mechanic into something different, and added a brand new "fighting style" that is actually not a fighting style, and... yeah. I think you should be aware of how very convoluted this "suggestion" is.
"Leader" as a role is an artificial rules construct, and the design team has been pretty clear in this talk and elsewhere that artificial rules constructs are not enough to give a class an identity of its own. Fiction first.
What is the difference between a warlord and a fighter who is also a strategic genius or a powerful personality with great magnetism in the fiction?
I'm not sure there is any. And even if there might be, it's certainly a lot weaker than the division between a thief and an assassin in the fiction.
There's no mechanical way of creating a fighter who is a strategic genius or powerful personality and have that actually reflected in the stats of the fighter at the moment. I guess you can say that your fighter is a strategic genius or a charismatic leader, but that means literally nothing. I guess you could go with your suggestion to make the fighter a 6 stat boosting class, but what happens then? Well, you lose the ability to make a surly and rude dwarven fighter, or a happy-go-lucky swordsman who is easily tricked and just not that bright. Are those archtypes in fiction? I'd say that yes they are. And your suggestion has just destroyed them as possible archtypes in D&D.
I suppose you could create some sort of ridiculous hydra-class abomination where you can select the stats boosted in combination with the mechanics that you want your fighter-warlord to have, and create some sort of weird hydra-based class where half your class is based around being the best martial combatant ever, and half your class is based around being an inspiring leader who directs troops in battle.
I don't see the point.
Rogue - Could be a thief, a swashbuckler, a thief-catcher, or an assassin. In all cases relies on their skill, careful choice of engagements, and is lightly armored.
Fighter - Could be an archer, a swordsman, an armored knight, or a guy with a spear and a grudge. In all cases relies on incredible mastery of their chosen weapon.
Warlord - Could be a strategic planner, a battlefield leader, a Sargent who manages troops or a commissar who inspires with rhetoric and threats. In all cases a battlefield combatant who exists to drive those around him or her onwards.
The most important part, since classes are more distinct mechanically than thematically (The Sorcerer is barely distinct from the Wizard thematically at all, the distinction is entirely mechanical) is that there's no clean way to incorporate the Warlord as part of the fighter without either making the fighter a class that is split in a way that no other class is or removing everything that makes a Warlord a Warlord. How do you plan to stick healing on a fighter exactly? Yeah.
To make Assassin part of a rogue, you can let them pick up proficiency with Poisons and Garrotes. Rogues can already sneak, so stealthy executions are in. Rogues already have a lot of skills, so the disguise skills necessary to get close to important nobles is already part of the class (Rogues have other reasons for having disguise anyway). Poison knowledge is a small subsystem (in every edition of D&D) and does not particularly define the class' identity.
In short, Assassins can be a type of Rogue because the Rogue archtype is broad enough to include them easily (also the designers have said they want a mechanically separate Assassin at some point). Warlords cannot be a type of fighter because making them a type of fighter threatens to either create a druid-like abomination (6/6 stat boost, really?) or just the single most confusing class since the Artificer -
and remember, Wizards wants the Fighter to be SIMPLE.