In order for that to be true, Warlord healing would have to include a rider that prevents it from working on dying targets.
Why?
Inspiring Word did not close wounds, it inspired. It was not magical. It's not like being KO'd makes you deaf, for instance, and people have reputedly been woken up even from comas by such things. It's entirely in keeping with genre - almost any more heroic or action genre, really, not even just fantasy.
A warlords inspiring word works just fine on a character at -20 hp, with two failed death saves, whose is also deaf, blind,
Deaf /and/ blind is getting to the point where I wouldn't object to a DM telling me, hey, you're just gonna hafta go over there and shake him or something...

Didn't come up in 7 years of running & playing that ed, though, and it's not something I'd do as a DM (tending to prioritize fun/drama over realism), so a pretty trivial corner case. 4e was not so much closer to perfection than every other edition that it didn't have it's little inconsistencies, either, while Inspiring Word was just a Close Burst, the various Commanding Presences each had different requirements to function. Some the ally had to see or hear you, others you had to see them, and so forth. :shrug:
20' away, a different type of life (myconid or xorn) and shares no languages with the Warlord.
It's unlikely you'd have an ally with whom you'd share no languages and couldn't communicate with in any way (no PC race has those issues, for instance), so no, that'd never come up.
The 'type of life' doesn't matter though, you could inspire a Shardmind for instance, or a Shardmind could be a Warlord, for that matter I played a Shardmind Warlord|Shaman in Lair Assault one season.
You seem very caught up on the idea that tactics must apply meta-game knowledge, and that seems to me to be contrary to your goals.
There is an important distinction between the abilities of players and those of their characters. It's easiest to see with physical stats and actions, but it's equally valid with any character ability. We don't insist that only players who can bench-press a buick can play half-orc barbarians, nor that a player who wants his character to disarm a trap try to defuse an actual bomb (a bomb squad trainer, say, to take danger off the table), by the same token, the 'tactics' that players engage in - focus fire, moving around an imaginary grid, and so forth - are not the same thing as the 'tactics' that a character meant to model a 'tactical genius' would be employing.
I'm sorry, I don't follow this at all. To clarify, I have as little interest in a debate on the nature of HP as I do in an edition war,
Then it doubly shouldn't be an issue.
but I am not a fan of purely abstract hit points.
I'm sorry you feel that way, but there's no problem with them in 5e. 5e includes HD and overnight recovery of all hps, that precludes it modeling any extremely serious injury from mere hp loss. You can recover all your hps, even from the verge of death, even with no aid whatsoever (assuming moderately lucky death saves), in as little as 2 hrs (1 hr to wake up with 1 hp, 1 to take a short rest). No wound heals in 2hrs, so any wounds you've taken don't impede recovery to full hps. You might still have wounds, but they're not preventing you from avoiding new ones.
There's no problem with changing that, of course, and the DMG has modules to do just that. Once you make natural healing 'realistically' slow, for instance, you can more easily visualize being dropped to 0 hps as being a very serious wound. If you're going to use such modules, you're unlikely to want to use an optional class like the Warlord, anyway - even if you do, it should be possible to simply not choose (or ban) anything that conflicts with it.
Just not an issue.
If HP damage reflects physical or even psychic injury, what could be more on concept then to apply first aid with a healing kit?
Nothing wrong with it, and anyone can do it, it's just not part of the warlord concept.
Since my reading of 5e's hit point system makes them less abstract than those of 4e,
You are free to think that, and 5e makes it easy to adjust the game (as above) to make it work more consistently with that sort of visualization, if you like.
I don't think healing, particularly healing capable of rousing someone on the edge of death, should be doable at a distance without supernatural power.
If that's what you choose to believe, then you simply don't do it. Seriously, there's no need for a 5e Warlord to have something like Inspiring Word hard-coded. Just as you could choose never to prepare Healing Word as a cleric or the DM could ban it, you could never pick Inspiring Word as a player or ban it as a DM. It's clearly not necessary to your vision of the class, unlike mine, so that's no issue for you. Similarly, I would be free to use it, since I don't have a problem with the commonplace genre trope it models working in D&D.
I admit I'm intrigued. While we seem to have differing approaches to the game, I would be very interested to see what you consider to be a good 5e take on the warlord class.
We can only hope something comes down the line. I look forward to analyzing it some day.
Are you familiar with the Hunter class and the Tactics feats chains from Iron Heroes?
Nope. Heard good things about Iron Heroes, but only in general terms. Like it's more 'low fantasy' and 'swords & sorcery' with a preponderance of non-magic-using PC classes.
Feat chains, though sure - in 5e, a chain tend to get bundled into a single feat. And, it wouldn't exactly be shocking to see some future material for 5e call back Iron Heroes.
Saying "I want a warlord" is on point. Getting into a multi-page back and forth over whether it should be there, or whether the concepts it represents should be there, isn't.
So folks should refrain from jumping in and challenging you wanting a warlord? That'd be a nice change.