(emphasis mine)
Right, this bit must have was added just to make 4E healing/math work.
...
Except, no. That's quote's Monte Cook in the 3E DMG. Long before max HP ever really related to 'healing' powers.
Hit points have always been the ability to both take physical punishment and to turn a greater blow into a lesser blow.
But in 4E, that is no longer the case. That's what I object to. Now, it's also skill, luck, and resolve as well because if you make it these additional elements, then shouting out words of courage and determination via Inspiring Word is a little less nonsensical. But, it can still be nonsensical.
If the Warlord shouts at the unconscious PC, he is restoring luck and resolve.
If the Warlord doesn't shout and nobody helps the unconscious PC, then the PC will probably bleed out and die.
Sorry, but it's just silly that the PC is dying, but the Warlord shouts and the PC is no longer dying. I'm glad it works for you and other people. But, it doesn't work for everyone (especially some of us that enjoy the fluff aspect of the combat powers) because it's not hit points anymore. In fact, this thread was written and we are discussing it here because it doesn't make sense to everyone.
I would have no problem if Inspiring Word handed out temporary hit points. That way, it isn't healing, but it is helping. To me, this is a better solution than the one proposed by the OP. That makes sense for someone shouting out encouragement.
But, I don't like it being healing. And, I've seen two players whose backgrounds said that their Warlord worships a deity and gets Inspiring Word from the deity. I wasn't DM in either of those two cases and it wasn't my PCs. In fact, we've had a grand total of two Warlords in our home game and both players did it. Some people just don't like it and no amount of rationale will make it all better. It's nonsensical.