You asked why it would be a problem. This is why it would be a problem.
Actually, I didn't ask that. I asked, specifically in response to epithet, why martial hit point restoration doesn't fit with the logic of the game; and neither he nor you have answered that.
What you've said is martial hit point restoration doesn't fit the logic of the game
when you choose to ignore part of the logic of the game.
That doesn't answer the question, it
changes the question; thus, moving the goal posts.
That's like saying the existence of a screwdriver doesn't make logical sense because I only use nuts and bolts, not screws; despite the fact that things you might need to work on are assumed to use both.
I however did show how a Warlords "healing" is consistent with the logic of the game; and nobody has provided a logical or factual refutation of that.
People are using hit points in a way that is 100% consistent with the RAW ("Hit points represent a combination of physical and mental durability, the will to live, and luck" - note the very first thing in that combination), interpreting them in a way that means that a desperate scream even from your closest love, the person who influences you the most, is not enough to get you to soldier on.
Actually, they are not using hit points 100% consistent with RAW if they ignore the mental durability, will to live, and luck aspects in favor of viewing hit points as only physical durability. That's not 100% at all.
A Warlord's "healing" works by restoring that mental durability. Since D&D doesn't differentiate between types of damage, the net effect is a restoration of Hit Points. And frankly, if that's the source of the dissonance, then why is a spell like
Cure Wounds not a source of dissonance either? It's name specifically designates wounds (physical durability) and not mental durability - yet nobody bats an eye at it healing hit points indiscriminately - including the unspoken mental ones.
Even in the real world a Warlord's abilities make sense. It's a fact that physiological shock (circulatory shock) can kill just as much as physical wounds; and maybe even more. Physiological shock can be caused by intense pain, and even extreme emotional shock, same as it can by traumatic wounds. Focusing the mind away from the mental aspect or away from the pain can reverse physiological shock. People can mentally hang on long enough for the body to regain homeostasis on its own; often only needing a short period of time in which to do so. More than likely it's the effect of epinephrine (adrenalin), but there's certainly an aspect of mind-over-matter that takes place. Once homeostasis is achieved, typically the body doesn't relapse unless something new occurs. That means it's generally permanent, and why temporary hit points are inconsistent with what a Warlord does.
Granted, the example of Sarah Connor and Kyle Reese is fictional, but it is grounded in reality. As someone who has actually been to war, who has seen such traumatic situations first hand, I can vouch for the veracity of this. Emergency personnel see this all the time. Even my wife, a hospice nurse, has seen this. She's lost count of the number of times she's seen a patient hang on long past what was believed possible, just in order to see a loved one before they pass.
I think the dissonance that many feel concerning this is more a failure of imagination and lack of experience, than it is one of illogic. A paradigm problem, not a logic problem.
Real world dying, represented by 0 hit points in D&D, is often a many-layered thing. Even dying of a gun-shot wound isn't completely black and white. Oft times an actual gun-shot wound isn't what kills, but is instead heart-attack or stroke, or even heart failure due to circulatory shock or hypovolemic shock.
But D&D hit points aren't that granular; and that's a good thing. The game would be far to complicated if it were. But just because the things that a Warlord "heals" aren't explicitly visible (though technically, no damage is explicitly visible), it doesn't mean they don't exist and don't have a profound effect.
My only point is to point out that this is why warlords actually healing HP without magic is a problem for some folks (and thus why it's probably overall a good thing that it was kept out of the PHB, even if we get it back through some means). You asked why, so that is why.
I disagree that this is a good thing. If design decisions are made based on some disregarding aspects of the game, that is a bad thing. If such considerations become a guiding element, then it's impossible for the game to ever improve. If such considerations had been made a guiding element during the design of 5E, there would be no 5E. There are always going to be aspects that don't jive with somebody; every D&D gamer has aspects of D&D that don't jive with them. Most simply ignore the parts that don't work for them. Excluding something because somebody might ignore it for this reason, is not a good thing.