Magic is what allows dragons to fly. Magic is not a crutch, it is the science of D&D. It is the physics of Faerun.
If you're claiming this, you're actually illustrating why "non-magical" Warlords are completely valid, ironically enough (I mean I assume that
wasn't your intention).
How so? Because you're illustrating the difference between "magic" and
magic.
Maybe a dragon does fly in D&D because of "magic", but it does not fly because of
magic. You cast dispel magic on a dragon, or flies through an anti-magic zone or whatever, and it just keeps on trucking. Is it realistic in a real-world sense? Obviously not, but as you say, the underlying physics of D&D are magical in a sense. Dragons couldn't fly, probably could barely move around IRL, but in it's not because they are casting spells or outwardly using magic in D&D that they can, or even that magic is coursing through their veins (note many other "impossible" things can fly in D&D too) - it's because they can.
Likewise with Warlords in D&D. Yes, maybe there is some underlying "magic" to the D&D world, that, despite not closing the wound, despite not making a beautiful glow, despite not involving an incantation or components or whatever, a Warlord can increase the remaining HP of a character, get them to keep fighting. That is indeed absolutely analogous to the "magic" which means a dragon can fly despite not working with RL laws of aerodynamics and so on (first person to mention bumble bees has to buy us all a beer). But it's very different to the outward facing, obvious magic that a wizard or cleric uses, and somewhat similar to the "magic" that allows a bard to increase how much everyone heals on a short rest, or the Second Wind available to Fighters (which is obvious non-magical healing but arguably "magical" in the sense you are describing).
I mean, re: Mearls, it's always been an issue that, like a number of game designers, and even some writers, even some very successful ones (looking at you, George Lucas), he doesn't really understand his own premises in some cases, and seemingly isn't capable of taking on board explanations from other people (which I find weird but Mearls has consistently shown an inability to even process what people are saying on stuff like this - and it's not uncommon to find people who just can't process certain ideas). This seems to me to be pretty easy stuff.
Even if you really want to be difficult about it you could just allow Warlords to trigger HD usage with massive bonuses and call it an adrenaline surge or something - maybe also let them grant HD on a similar basis. Is that "magical"? Perhaps. But it's not
magic.