Here we get into yet another place where it all depends on the way you choose to narrate. Since 4e did away with nonlethal damage, the only time you fall unconscious is when your character is "dying". Once your character's hit points drop below 0, does that always mean that your character is literally dying? If your character recovers and gets back up, then he wasn't *really* dying, he was merely walloped pretty good and knocked unconscious for a bit. For me, it isn't a matter of suspension of disbelief when the warlord brings someone back from the brink of death.
But the warlord
decides whether the guy was really dying or not. If the warlord yells at him to get up, then he wasn't really dying. If the warlord doesn't yell at him, then he
was really dying and dies. And the warlord is acting as if he knows this. That's where the suspension-of-disbelief issue comes in.
A lot of us are very uncomfortable with a system that requires players to make tactical choices that their characters have no in-game basis for making. Going by your explanation, the warlord has no idea in-game that he can affect his fallen friend's survival chances by yelling at him. But he's making decisions as if he
did know that. Instead of using his minor action for something else, he uses it for Inspiring Word. If he's more than 25 feet away, he moves to get closer so he's within range, possibly even provoking opportunity attacks.
Why is he doing those things? Out of game, he's doing them because his player knows the fighter is on his third death save. But in-game, his player has to make up some random explanation. If I'm that player, the act of making up that random explanation smashes my suspension of disbelief into little bits; it's a forceful reminder that my character is just a game construct. I'm not trying to get into my character's head and make the decisions he would make. I'm making the decisions
I would make and then justifying them after the fact.
This is why I prefer my "will to live" explanation. It's not perfect, but it doesn't require this kind of separation of player and character motivations.
(I would really like a system that imposes long-term penalties for going below zero hit points. That would both address my suspension-of-disbelief problems, and negate the problem of players holding their healing in reserve until somebody hits zero, which is something I see a lot of.)