But as long as we're talking about "quibbles of fluff" how about the "and it must be non-magical healing" fluff? I have as much trouble understanding that requirement...in a game full of magic and fantasy...as you have understanding why I feel inspirational healing impinges on my own player agency.
A "game full of magic and fantasy" often leads people to two differing opinions about the nature of their ubiquity, and that rub of your point lies in how one reads "full of." I am increasingly coming of the opinion that a big part of this discussion entails these two diametrically opposed, but equally-supported and valid, approaches to playing D&D over the past five editions. Some people prefer one or the other. Some people are fine with both.
If "everything is magic," which is a position that many take regarding D&D and its classes - including barbarians, rogues, fighters, etc. - then yes, the warlord and its abilities would naturally, and inescapably, be magical too. But if "not everything is magic," which is a position that many others take regarding D&D and its classes - including barbarians, rogues, fighters, etc. - then this is the side of the fence that many warlord fans would prefer the warlord to fall. But why is that the case? Why not the other side?
You know, I don't know about you, but what I do know is that when everything in the game is "full of magic and fantasy," sometimes people want to play as that non-magical fighter, rogue, ranger, barbarian, and, yes, warlord who persevere against the odds despite their lack of magic. (There's a reason why there is often a high demand for non-magical rangers.) And to that end, the warlord was a particularly notable option that empowered many such players in that latter category to not only play within that (fairly popular) fantasy conceit, but to do so in a way that differed from the other 'non-magical' classes. In a way that was oriented more towards a mechanically-backed support role, which was a role that D&D traditionally reserved for mostly casters, particularly divine casters and bards. So to appreciate this "quibble of fluff" about non-magical warlords "in a game full of magic and fantasy," I think it helps to sympathize more broadly with that desire some players have of being those characters in that world "full of magic and fantasy" where they are counted among "not everything is magical."
But we can also go back to the earlier approach to D&D gaming then in which "everything is magic." If we accept the idea that "everything is magic," then at least one point of contention becomes clear within D&D: "Everything is magic, but apparently things like 'anti-magic' are ineffective against the 'subtle magic' of mundane fighters (et al)." If we take "anti-magic" abilities to be oriented against 'spells' (i.e. 'casters') as opposed to strictly magic in a broader sense - the one in which fighters, rogues, barbarians, or even a bard's inspiration - then we are effectively creating multiple types, tiers, or classes of 'magic.' There then becomes 'magic' affected by 'anti-magic' (casters casting spells and some of their associated abilities), and 'magic' that is not affected (fighting men et al). It should become clear then, when bringing the warlord back into the discussion, that again it is this latter category that most warlord fans would prefer if one prefers the conceit that "everything is magic." It's why playing a bard or cleric still does not work for many warlord fans; it still involves casting spells that belong to this 'other' class or brand of magic. It's a magic that exists outside of this other class of magic. If "everything is magic," then the warlord and its abilities should exist within that ambiguous space - wherever that position lies - that allows that subtle magic of fighters, barbarians, rogues, bardic inspiration, a paladin's lay on hands, etc. to exist apart from the effects of anti-magic.
These two approaches often have two completely different takes on the warlord and their abilities. It feels as if the various sides - because to be clear here, there is more than two - are talking past each other and failing to acknowledge
both these approaches to the ubiquity of magic and its relation to the meaning of a warlord in a "game full of magic and fantasy." This is why I and others (both pro-warlord and otherwise anti-warlord), here and elsewhere, have proposed that the flavor text for the source or nature of the warlord's abilities should exist within that extraordinary, semi-mystical ambiguity that permits people to use warlords for either the "everything is magic" or "not everything is magic" approach to the warlord. Demanding that the warlord should adopt one hardline flavor text over another hardline flavor text effectively screws everyone in a way that is not conducive to a long history D&D that allows both "everything is magic" and "not everything is magic" as viable campaign options within the parameters of its rules and flavor text.