Quite good, I like it.
There's one thing tho, and it may sound counter-intuitive: I dont think all the ''warlords'' (this is one of the reason I dont like the name) features needs to be combat-focused. I think that a martial support class should be able to help as well in combat than out of it. I even think warlord should feature abilities that remove or diminish the need to enter combat. Something like setting ambush, avoiding encounters, using cha/int/wis skills in combat instead of fighting rolls. This is why I liked the idea that someone said on one warlord thread or another, that the ''spell-less ranger'' archetype could be created in a warlord class quite easily.
It may seems weird, but I think a ''warlord'' should be the class that masters the exploration and social pillars instead of the combat one.
They're going to need a great deal more than that to catch up with "also being a full caster with a strong spell selection". I mean, the L1 ability, for example, has no bonus and a pretty harsh limitation (perhaps INT mod + half your level would make more sense - lower cap, higher base). A Bard has +1d6 on ALL HD usage in rests with no limits whatsoever, and that's on top of being able to cast actual healing spells. Sure it's out of combat only, but for the Bard, it's just a generic minor ability, whereas for the Warlord it's a core part of what he does.
I do like the general direction you're going in though. Just don't forget all the OTHER spells Clerics and Bards get alongside the direct buffs.
There's plenty of levels left to add other stuff. Like alternate uses for reactions (reroll when an enemy hits), sub-class skills (ranger group can move overland faster while using stealth), and such. That's just a skeleton. It still needs some meat an skin.
Also keep in mind, help action super-advantage will work for skills as well.
Also, I didn't want too much healing at level 1. Spending a hit die is enough to get off the ground.
Level 1: As a bonus action, you can allow someone to spend a hit die. You can do this a number of times equal to your class level per long rest.
Sub-class: (bonus proficency / expertise)
i.e. rangers get expertise in survival. Medics get expertise in medicine. warlords get weapon and armor. diplomats get persuasion
Level 2: when you take the help action, the target can roll 3 d20s instead of 2.
Level 3: Sub-class: (you can use your action to...
i.e. rangers can allow everyone to make a stealth check. Medics get +Int to healing. warlord can attack and help in the same action. Diplomats can charm.)
Level 5: As a reaction when someone misses an attack or fails a skill check, you can let them reroll the attack or check.
Level 6: Sub-class (you can use your reaction to...
i.e. rangers can move and hide as a reaction. medics and move and heal if someone drops. warlords can use the help action when they make an OA.).
... (more stuff, like initiative, reaction to avoid OAs, out of combat sub-class skill stuff...)
Level 11: you gain an extra reaction.
... (slightly more fantastic stuff, reaction to reroll failed a save with disadvantage, full action grant 1/rest, prevent a failed spell from using resources, more sub-class stuff).
Level 20: When you use the help action, the target d20 roll cannot be less than a 8 + your Int modifier.
