The fighter has self healing of real HP.That’s def part of it. But I don’t particularly have a problem with martial healing personally. It’s just I recognize it is against enough peoples play style that it’s typically best avoided.
That said, I think temp hp is actually better conceptually for warlords than healing. Temp hp applies to fresh ally’s. Something a leader should be inspiring to fight better.
Honestly the 2023 5e community is different from the 2013 community who filled out the DNDN playtest.
I think today's community, the one that is majority under age 40 and about 25% newcomers, have not problem with warlords healing HP.