And we've come full circle back to gamist mechanics devoid of any meaning in the game world. You're one step away from "1[W]+Str and target is pulled 3 squares".
"Martial", as I was defining it, was classes that are generally good at combat. Defined as "proficient in martial weapons, most if not all armors, d10 or greater HD, and two attacks at 5th level". This category covers the Fighter, Barbarian, Ranger, and Paladin, as those classes can more or less swap for one another in the role of combat while bringing different secondary abilities to the table.
Nonmagical, as you are describing, is somewhat limited: The fighter is decisively nonmagical, as is the rogue (barring certain subclasses).The Barbarian is fairly nonmagical, though you can argue rage can be viewed as a supernatural ability beyond mere anger issues, and the monk has ki as a source of supernatural exploits. Rangers and Paladins have spells and divine powers.
Unfortunately, that is a rather small box. The best place to look for outside the box in 5e is Battlemaster (as they can grant extra attacks, temp hp, and a variety of tactical attack). My concept is to expand the battlemaster into a full class and use it as a guide to develop more powerful effects. But effects that, at most, would be on par with 5th level magic tops (and far more narrowly defined than anything a current caster can use). I think there is design space there that won't cause outright rejection by the D&D community at large.
This to me doesn't say "I want a class that is good at tactics and support" and says "I want a class that I can use to re-write D&D to be low/no magic." That to me is effectively a non-issue. The warlord should work not as a cleric replacement for no-magic games, but as a unique element on the D&D Multiverse.
I'm sorry, but the concerns of "low/no magic D&D" rank right up there with the lack of modern technology like cell-phones and cars in the list of Things I Don't Think Should Matter When Designing for Dungeons & Dragons.
Only because you refuse to compromise. The designers, starting from the Battlemaster up to today's Warlord HFH, view the Warlord as basically worth a fighter subclass. A good compromise is a full class that is a hybrid of a fighter and a support class who isn't required to check all the boxes of a support caster nor is he bound by the constraints of a fighter subclass.
Put another way, would a paladin-like official warlord be better than the no-warlord you have right now?
Many of the effects that the "band-aid" support classes are there for are unfortunately magical, and have magical solutions. I'd say this is because D&D is a highly magical game, as evidence by in every edition magical/caster-classes outnumber non-caster classes by a large margin. To me, a martial "support class" changes many assumptions about the game at the fundemental level. That's not necessarily a bad thing, but I feel its beyond the scope of D&D and what the designers have laid out for it. It really sounds like a warlord is a better fit for a non-magical game like The One Ring d20 than D&D.
As stated above, the ranger in 5e is a lot closer to fighter than rogue these days.
In 5e, a paladin is a warrior who can support his allies with healing (lay on hands), defensive buffs (auras), and spells (bless). A warlord could easily fill the same role as a warrior who gets healing (inspiring word), defensive buffs (command zones) and spells (gambits).
The issue with the "warlord as contrroller" gets into the nasty mess of disassociated mechanics and "martial mind control" that was a huge sticking point of 4e. I'd rather avoid those issues and work in the design space hollowed out by 5e to try to ease those concepts in rather than throttle the gas and bring back Come and Get it.
4e made the low/no magic game possible by beating magic to death with a stick. 4e magic and 5e magic are light-years apart, and I'm afraid there isn't enough ways to make up that distance without grinding magic back down to "attack + riders" and rituals again.
No, its a HUGE thing to ask.
4e is still viewed as the poster-child of everything people hated about 4e; disassociated mechanics, powers, source/role grid-filling, encounter/daily martial powers, etc. WotC has to understand that any warlord that brings those elements back to 5e in a large, unchecked way is going to cause huge amounts of grief. A martial character tossing "nonmagical" equivalents to Foresight is going to grind more gears than it doesn't. Simply put, there isn't a big enough market to warrant its design and development, esp since WotC has been slow and cautious on introducing new classes (the artificer and mystic are still in development hell). Even my paladin-looking warlord, while probably less problematic than a "casterless caster" is still a large amound of design and development to satisfy a rather small minority of the fanbase.
Moral the story: the wilder the design goals of a warlord, the less likely it will ever see creation. I tried to shoot for a compromise by making him partially a warrior, based around an existing mechanic (superiority dice), and adding in as much of the 4e class as I could given 5e's definitions of magic and mundane power.
A nonmagical cleric has literally no chance in hell. A battlemaster+ as a full 20 level class, though just might.