Warlords are specifically non-magical. That's the entire point. It's only "magical" if you take the presumption that HP=meat. Otherwise, it works perfectly fine.
While Warlord healing is consistent with what Gygax says about hit points in the AD&D DMG, since
most of the ways to regain hit points quickly are magical, it's not entirely unfair for there to be a disconnect in people's minds. Also, there are circumstances where hit point loss almost has to involve actually being hit and physically harmed (Micah Sweet often brings up attacks that poison you, for example).
We've been trained to consider a "hit" that inflicts damage to be physical, even if it makes zero sense for a 10th level Fighter to be able to shrug off 5-6 sword blows that would take out his level 1 compatriot.
While versions of D&D prior to 4e did have non-magical sources of rapid healing, they were rare and didn't stand out as much. I've had people tell me the main problem they have is a "martial" healing others, not themselves (like when I bring up 5e's Second Wind...despite the fact the Banneret exists, lol).
Until the game stops calling the primary healing spell "cure wounds", and solidifies what hit points are and how they function, you're always going to have this disconnect, with some ridiculing the idea of "shouting hands on" and other ludicrous hyperbole.
The best way to handle this, IMO, would be have future Warlords hand out temporary hit points instead of actual healing for the most part (with an exception for reviving a dying character by ordering them to do so- something that has been documented- or simply commanding them to "don't you die on me, soldier!", which happens so often in media that it's also probably more acceptable). If you're going to call yourself a "martial", then rather than seem supernatural, it's best to model the extraordinary feats of your average action hero.
If we can accept an aging Liam Neeson or Denzel Washington as being a one-man army and not be immediately taken out of the experience, then that's the level of "fantasy" that should be modeled after.
I'm saying this as someone who had no problems with the 3.5 Marshal or the 4e Warlord, but after years of hearing the detractors naysay it, I'm willing to make concessions in order to maintain the "flavor" of what it means to be a purely magical character vs. someone who merely exists in a magical world.
This won't completely solve the problem, however, you're still going to have people who reject the idea of a class "ordering" people around (why the Mastermind Rogue slips past this criticism is anyone's guess), and similarly reject any sort of martial "taunting" system in order to allow for a functional defender* (which has been going on since at least AD&D's Kender).
*Granted, this just one way to accomplish the task. The 3.5 Crusader and Warblade could assume a stance that imposed a hefty penalty to attack anyone but them (a
14th level ability in 5e), and while 5e seems incredibly loath to do so, just giving classes the ability to perform multiple opportunity attacks (an
18th level ability, outside of a Fighting Style that never left Unearthed Arcana) at reasonable levels would go a long way.
