But should a motivational speech really wake someone up from being beaten unconscious?
If hit points are mainly about luck, grit, determination, willpower, and the urge to keep fighting, and as a result being at 0 hp doesn't necessarily mean you've suffered a lethal wound, then yeah, it should.
If being at 0 hp means you've basically suffered a lethal wound, no, it shouldn't.
Different tables will have different answers to what is happening in the fiction when you reach 0 hp. A lot of warlord fans are fond of the former, which means, of course, that there is absolutely nothing wrong with being woken up from unconsciousness by a desperate shout from an ally who is encouraging to others by their very presence. Indeed, it makes perfect sense.
Any kind of warlord that would be consistent with the latter might not be enough warlord to please some warlord fans, I'm afraid, because of that requirement.