I think you over-estimate a caster's versatility. A 5e cleric does have about 15 spells to choose from at a given level (not much more than your dozen). Wizards only get a guarantee of knowing two spells for every level they gain. Furthermore, a fighter probably IS the highest-damage-output at most levels. Compare a wizard who fire bolts (1d10) to a fighter who longbows (1d8+DEX bonus), or even better a fighter who Mauls (2d6+STR bonus).
While I admit that much of my experience with the 5e Fighter is academic (as I have yet to see anything that would make me
want to play that class), I have to wonder if the frequently-cited Fighter damage numbers are
actually as good as people talk about. Of course, many times, I've tried to actually compare a "pure damage" Fighter with a "pure damage" Paladin (really I should also consider the Barbarian, I just haven't because Pally is so easy to calculate)--and pretty much every time, I received a dismissal for "theorycrafting." The numbers I have crunched, though, don't really reflect the Fighter being a breakaway damage-dealer--more like slightly leading the pack.
In fact, part of the reason a battlemaster is in some ways a sub-par warlord is that the battlemaster still makes a lot of attacks themselves, and gets their power from that, while warlord fans seem to want a class who gets a lot of their damage potential from giving extra actions to their allies. The battlemaster gives actions to allies as a cherry on top of being an engine of destruction themselves, rather than giving actions to allies as a base part of how they contribute to damage/defense.
[sblock=Semi-digression]This I can definitely agree with. People focus a lot on the "Lazy/Princess" build of the Warlord, but the truth is that
all of the Warlord builds were pretty keen on one specific thing:
Contributing by making others' contributions better. Sometimes, that manifested in the Lazy way: "you, attack this brute, and go for the armpit where the armor's weak!", so the ally gets an extra attack with some damage added on top. Other times--more commonly, I'd argue--it manifested as quite literally
leading the attack: "have at you, you foul miscreant--and taste my comrades' steel as well!", so that
future attacks made against that target get some kind of benefit. Other manifestations could also work, which were fluffed (while I recognize the ease of refluffing in 4e) as exploiting weaknesses, shoring up defenses, or other forms of Natural Battle Enhancement.
The Warlord could totally heal, but again it was more about being..."facilitative" than about being "reconstructive," if that makes sense. An ally that remains in the fight is extra damage, extra controlled space, additional flanking ability, another body to take some heat off the squishies, etc. Some Warlords are even facilitative by taking risks; Bravura (a Cha-based Warlord) was all about risking or even accepting downsides (like a penalty to an ally's, or your own, defenses) in order to do something dramatic and rewarding IF you pull it off. Others (e.g. Tactical) were facilitative primarily by accelerating the pace of combat, particularly by improving Initiative, or by making the party highly maneuverable (a less-useful thing in 5e, since everyone can move-attack-move-attack etc., but that doesn't mean it couldn't be worked with).
But the big thing Warlords really
weren't was being particularly powerful on their own. Being a "Striker" in 4e was essentially the one thing a Warlord
definitely couldn't do both well and consistently. Which is something of a problem, because the base chassis of the 5e Fighter IS definitely a damage machine--being the only class that gets four attacks (though, in practice, most people will only see 2-3) kinda forces the Fighter to be
at the very least competent at personal damage-dealing.
For a 5e rendition of the Warlord vis-a-vis the Battlemaster, the baked-in Extra Attack of the Fighter, plus all the personal damage-improvement (or personal hit-improvement) from
nearly all Maneuvers, is somewhat an issue. (Of the four maneuvers which don't improve your own damage or hit, two are personal defense boosters--Parry and Evasive Footwork--so only Commander's Strike and Rally are "facilitative" in the way a 4e Warlord was.) It's essentially unavoidable that the Battlemaster will be a major damage machine; the mechanics really don't contribute to a player focusing on how to leverage their allies' strengths on the battlefield, which the 4e Warlord
had to do in order to meaningfully contribute to combat. Healing--leveraging their allies' HP pools effectively--was a part of that, but could be made optional. Making the
whole thing, the whole "facilitative" structure, optional...that's not so doable, IMO, without making the class something fundamentally other than a Warlord.[/sblock]
Long story short: I completely agree that the naturally good-to-great offensive (and to a lesser extent, defensive) capabilities of the Fighter chassis, and particularly the Battlemaster archetype, conflict with the desired execution of a "ported" Warlord.