Because if they cannot heal, they aren't actually an effective support in a game designed around ablating away HP.But why?
Why is “can do cleric stuff” vital to the spirit of the class? That is exactly the thing I don’t understand.
It’s not like the warlord was performing miracles of physical healing in the fiction, so why does it matter, as long as they are inspiring their fellows to go beyond what would otherwise be their limits while fighting alongside them?
Like... that's literally it. It needs that because if you cannot heal, you literally cannot play support properly in D&D. You might have a smattering of buffs, but you cannot truly be the support character other characters rely on.
THP are completely unacceptable as a substitute. Either they don't actually match 100% of incoming damage, which means eventually the Warlord is letting the team down because they die and she can't do a damn thing about it, or they DO actually match 100% (or more) of incoming damage and thus the Warlord is stupidly broken because she literally makes her allies completely invulnerable. Neither of these situations is acceptable. Therefore, whatever other things she can do, whatever other emphasis she may have, she MUST, absolutely and unequivocally must be able to do SOME healing.
It doesn't have to be a lot. But it needs to be there. Why else do you think the PDK, an obvious (if extremely poor) attempt at making an actual Warlord in 5e, got the ability to share Second Wind with other people?
A core part of the spirit of the Warlord is that you are a real, serious, competent support for all parts of combat. You can be relied upon to keep the team going through the stuff they expect to face. Healing damage taken is a critical part of that, particularly in 5e's "big bags of HP" design, where HP inflation is one of the only metrics of player advancement/increased difficulty at higher levels.