Hit points are really just plot armor. Any damage characters take is "just a flesh wound". You have to use 80's action movie logic. In our reality, healing magic would be amazing, but in a D&D world it's nothing special- which is an intentional design decision, since WotC doesn't want people thinking they have to play a character who heals damage.
This is why 50gp gets you a Resident Evil-style "first aid spray" that characters can even make themselves (...with 50gp of green and red herbs, presumably)!
Magical healing hasn't been special or unique for 20 years now, heck, in 3.5 Fast Healing not only wasn't considered magical, even the Fighter could get it!
But some people still want "realistic" healing in their fantasy game, so that an all martial party has to take two weeks to heal up from one battle...even though they have to warp and bend the game's rules in order to make it come about (altering when you can take a rest)!
Which, unless you're playing in a "West Marches" style game, where you switch to different characters, just leaves you with large amounts of downtime that I presume are skipped over between sessions anyways, all to preserve one's sense that this isn't just a make believe game.
And if you're going to alter the rules that much, you shouldn't care if the Warlord can heal, because you'd just alter or ban them for your games anyways.
This is really my main issue with people who don't like the Warlord concept (including predecessors like the Marshal)- rather than treat it as what it is, an option that can be altered or discarded, their arguments come down to "the Warlord
exists, therefore it must be condemned".
I don't care for the new classes from 1e's Unearthed Arcana. Guess what? Were I to run an AD&D game, I could just ignore them. Oh sure, I might have that one guy who is upset he can't play one (there's always one, any time you remove an option), but it won't affect the game itself.
What really annoys me, though, even if WotC decided the Warlord is a bad fit for the PHB, they had 8 years to drop him into a supplement. Instead they made a bunch of "mini-Warlords", like the Battlemaster, Purple Dragon Knight (who totally can heal people without magic, btw!), Mastermind, etc.. As if they know the concept is viable, but they're terrified to commit to it.
Meanwhile we have the Monk, which is a much more tortured concept that no gaming company has ever gotten right, and which rumors persist that someone in WotC actually
despises.
I don't get it, at all.