You're right. Warlord don't need to restore real hit points just because they did in 4E. Warlord Inspiration needs to restore real Hit Points because it's part of the Warlord concept - A fundamental part of the concept as expressed by Warlord fans.
No, its part of the /4e/ concept of a warlord. The 3e marshal didn't have healing. The Dragonlance Noble didn't have it. The Star Wars Noble (d20 or Saga) didn't have it. No 3e Prestige class class that focused on leadership (like Legendary Leader from HoBattle, Purple Dragon Knight from the FRCS, Tactical Soldier from the Minis HB, or any Pathfinder ones like Golden Legionnaire and Noble Scion). Out of all those examples, (some produced AFTER 4e in Paizo's case) NONE have hp restoration via nonmagical means. The 4e warlord is in fact in the minority as far as leaders go.
Its like the Artificer. The Eberron 3.5 artificer was good at buffing (via augments to magical items or constructs/warforged) so it made sense to make him a 4e Leader (the role focused on buffing). However, the artificer wasn't a healer (he had no natural healing spells save those that worked on warforged and constructs alone), and in 4e, Buffer = Leader = Healer. So the artificer got magical healing potions they could make rain on allies to give them healing surges twice per combat, because that's what Leaders do. Now, would you say that healing is a core part of the artificer
concept? That any 5e artificer that comes out MUST have healing potions (at range no less) or they are a failure?
So, now it's your turn to answer a question: Why are you against the Warlord providing real, physical, Hit Point recovery through inspiration? Is it personal preference? Mechanical considerations? Or other?
Well, I'll answer this two ways:
I have no problem with a warlord rousing an ally rendered unconscious from battle using just 6-seconds worth of words from 10+ feet away IF that power was somehow "magical". Not a spell, but a supernatural power like a paladin's lay on hands or a bard's song of healing. If you will concede "nonmagical", I'll concede everything else.
OR
If you demand "nonmagical", then I must demand limits on its use.
A fighter can only use second wind when he's conscious (since its a bonus action to use) so no healing from unconscious (besides, unconscious condition says that the afflicted "is unaware of its surroundings", so it can't even hear your little pep talk).
Secondly, a fighter can only use SW once per rest, since it draws on "a limited well of stamina", and since the warlord isn't giving them new stamina (via magic) but triggering them to draw on such a similar well, I'd argue the warlord could only use this ability on a target once per short rest each. Further, since the warlord is speaking words of encouragement, the recipient must be able to hear them and understand them. Therefore, the healing wouldn't work if the target was deafened or didn't share a language (the latter isn't a concern among PCs who share Common as a lingua franca).
Lastly, while Healing Word is a bonus action to cast, its 1/2 as effective as Cure Wounds (which is both touch and an action). A warlord's healing should be similar; a small amount quickly from afar or a larger amount slower from adjacent. And since neither scale automatically, neither should a warlords (though some mechanic, like dice or warlord points could be used to augment that healed.)
Essentially, a warlord's nonmagical healing becomes "trigger a second wind-like ability in a target, once per target per short rest, as long as they aren't unconscious and can hear and understand you" which works just fine for hp recovery. He's still limited as a full healer (since he can't get Lesser/Greater Restoration) but he's at least recovering free hp for the party like a cleric can.
Work for you?