Of course first you have to assume that the character's spirits are down and that they have lost hope. Which conflicts with a character who is an eternal optimist and never let's things get him down.
The idea of the 4e fluff text being quoted is that buoying the ally's spirits restores hps. If you have some hope left, you have some hps left. In that context, the 'eternal optimist' you posit would have to have infinite hps. So, sure, if you talked the DM into giving you infinite hps, you would never need an inspiring word. Seems consistent enough. Also seems like you want to play very extreme, one-dimensional characters, for the sake of bolstering a position you've already conceded, above.
It always conflicts with something. That's part of the problem.
No, you can always make up something that conflicts, you are inventing a problem where none exists.
Look, you've already acknowledged that you have no problem with the Warlord being an optional addition to the game. There is no possible way it could be anything but an optional addition to the game, so what are you even trying to accomplish with this line of non-reasoning?
And a lot of these examples are of characters focusing their training in certain areas, and can be answered by multi-classing. Of course you can always multi-class into Warlord, so I guess that's a wash. As long as you don't mind having to take levels in Warlord in order to be an inspiring figure.
That's an aspect of a class-based system. If you want to mix capabilities of different classes, you do some MCing. 5e takes it a bit further with background and feats that partake of most classes' shticks. You can be a something/rogue, or you can just take the criminal background. You can be a something/Cleric or just take the Acolyte background and maybe Magic Initiate feat. You can be a something/warlord or take the Soldier background for rank and maybe the Inspiring Leader feat.
Really, the maybe-a-little-bit-warlordy stuff is already in place. Backgrounds, a feat or two, a couple of SCAG sub-classes. It's just the full class that's needed.
You can think of the magical Bless spell like taking performance enhancing drugs, but without any negative side effects. You are pumped up, you are energized. The source doesn't matter, Bahamut or Wee Jas, it has the same effect.
But what if you're an 'eternal optimist' who's always pumped up & energize? Or you're Eyore and /never/ pumped up & energized? The magic overrides that (no saving throw, no opt-out), taking away your player agency in that instance.
If it's inspiration, OTOH, you can always decline it, so you clearly retain your agency.
Here's another Tolkien-related exercise which may throw some light on the discussion. Imagine a party of PCs who relate to each other much as the Fellowship does (large group!). Aragorn has "Ranger" on his character sheet and Gandalf has "Wizard." Now, imagine that any of the others has "Warlord" written down. How would inspiration in the party work then?
Imagine any of them having 'Cleric' written on their sheet and it'll make even less sense.
And if Aragorn doesn't have "Warlord" on his character sheet, what's happening at the table between his player and Boromir's in that Moria scene quoted earlier?
Maybe he has a level of Warlord, maybe he has a feat that gives him one warlordly trick, kinda like Martial Adept gives you a Battlemaster trick. Maybe the fact that literary characters preceded the class system means they don't have to conform to it. Instead, the system needs to model them, and a Warlord would help D&D do so a lot better.
No, but it is described as magic elsewhere, and the article says that being described as magic is one of the tests to apply.
No, the quotes you mention don't describe the Bard's inspiration power as magical, they describe the bard as using magic - which he does: he casts spells.
So what's the difference between drawing on the power and magic? When we talk about wanting a non-magical support class, we want to be able to produce effects without reliance on spell slots, without those effects being dispelled by dispel magic, or negated by anti-magic zones, or coming with any religious or arcane baggage like demanding deities, pact-patrons watching over your shoulder, bloodlines, colleges, or cabals of wizards.
If you need "because magic" to justify stuff, cool. Have at.
That's another issue that was hashed out in this sub-forum, already. If you need to believe that only magic heals, or only magic inspires, or whatever, you /can/, just leave the door open to that same sort of (EX) power or magic-of-the-universe explanations as Bardic Inspiration, Dragon Breath, or what-have-you. Rationalize it.
See, I don't think that mundane non-magical words from a mundane non-magical person work that way unless I have a particular relationship with that person.
There's a lot of room between non-magical and outright mundane. PCs in an heroic fantasy game may or may not use magic, but they're not likely mundane. Extraordinary, but not supernatural, might be a better way of thinking about it. Bless or Aid uses a supernatural agency (divine power) to force a specific feeling on a target whether it makes any sense for him to feel it or not. Inspiration can produce similar feelings, and it's a RL (mundane) phenomenon. In the context of heroic fantasy, you might very well have an Extraordinary individual who does something that's not supernatural, but does it to a degree or with a consistency or a disregard for mitigating factors that far exceeds the mundane. Just having more than a handful of hps is an example of such extraordinary ability: a D&D character can enter into quite a lot of danger with virtual certainty of survival (without lasting consequences) that'd be quite impossible in a mundane context.