However, there is a downside to my idea that's worth noting. Multiclassing. When you take two caster classes, you don't get independent spell progressions- you continue to progress in spell slots according to a set progression- 1/3 for Subclass casters, 1/2 for Paladin, Ranger, and Artificer, 1:1 for Cleric, Wizard, Sorcerer, Bard, and 1/???? for Warlock*.
*I am one of those heretics who hates the Warlock's design so I've never played one. Granted, this is mostly because I came into 5e through Adventure League, where you rarely get a short rest, so being a 1st level Warlock is basically "cast 1 spell, then spam Eldritch Blast". Sure, that's only one less spell than the full casters, but it's still half the spell slots.
Now if I was a 1st-level War Cleric and I multiclassed into, say, Wizard, I get another spell slot, more cantrips, and access to 1st-level Wizard spells.
If I was a 1st-level War Cleric and I multiclassed into my proposed Warlord, since it's not magical, I now have the rough equivalent of 5 spell slots! That could be a huge boost at low levels, and yes, the "Warlord dip" could become a thing.
And I honestly don't see a way around that, because to my mind, shoving features that let you feel like you're playing your class behind a level wall isn't a great experience. You just have to hope that not having level 3 spells by level 5 (and so on) is a large enough price to pay.
And note, I haven't even gotten into what Warlord 2 would bring (my initial idea is something like how Pathfinder 1's Cavalier could loan out Teamwork Feats to their allies).
It's a quandary, to say the least- it's not that you can't have a level 1 Warlord. It's just that the game is balanced around "casters work this way, non-casters work that way" and bridging the gap is going to take more work than just slapping 20 levels of a class and a couple subclasses together and dropping them on the game.