The problem was, they decided each party needed a Leader. Leaders buff allies (in various ways) and healed. If the Warlord was a Leader that couldn't heal, they wouldn't be able to be slotted in in lieu of a Cleric, Bard, Ardent, or what have you. So the Warlord had to heal, and given that hit points have never been fully meat, they decided it wasn't so much healing as the Warlord's leadership spurring you onto greater heights.I completely agree, there will always be corner cases, but as clearly pointed out by the 5e design, the system is not there to deal with corner cases, it's the DM's job (otherwise the system becomes either unmanageably complex or too restrictive) and second hit points and healing are flexible enough in their concept to make corner cases rare.
For me, the problem with the 4e warlord is that it made suspension-of-disbelief-breaking "corner cases" the standard because you did not have multiple healers in your party, which meant that most of your healing was of that "suspension-of-disbelief-breaking" kind.
Which was all the more annoying since I liked the concept of the class, and it could have been solved by more buffing rather than through actual healing, which was for me lazy design.
Moreover, I enjoyed the class as long as I enjoyed the formality of combat in 4e, which did not last that long, and when I got tired of my powers just pushing miniatures around on a grid, I also got tired of that version of the warlord.
Now, the best warlord in our current campaigns are my Paladin in Odyssey of the Dragonlords because she is a natural commander although not relying on class powers, and a warlock in my Avernus campaign because she built on her pact with Mephistopheles to start commanding infernal and other troops and using her cunning, charisma and guile to keep them in check and make them do what she wants, both on the battlefield and in negotiations. These are real warlords, until I resuscitate the Wheel of Time RPG and get real commanders again.
What would have fit the narrative better would be a mechanic like the 3e Bard's Inspire Greatness, that actually temporarily gives you more Hit Dice and lets you function (more or less) as if you magically had 2 levels of Warrior bolted onto your character.
But, in a shocking move for 4e design, they went for the simpler, less complicated solution.