Hiya,
El Mahdi: Thanks for putting so much thought into this!
Moonsong: I haven't looked at the Noble class yet in EN5ider, but I'll be sure to check it out.
Regarding El Mahdi's suggested class powers: I kind of agree with others that a lot of them seem overcomplicated.
Command Presence: Asking for party consensus will slow the game. I'm generally not a fan of integrating Advantage/Disadvantage into class powers, but it's appropriate for the Warlord if limited somehow. That's why I suggested a fighting style or maneuver early in the thread where the Warlord could grant advantage to an ally if they were also at risk themselves (like the pack tactics monster power). If you want to also support a "lead from the back" warlord I would make it start at 1 use/short rest, and possibly increase it to 2/short rest at an otherwise dead level. But I personally prefer a pack tactics version.
Inspiring Word, Rally the Troops: I'm a big fan of martial healing in whack-a-mole combats, where the HPs clearly don't represent meat but luck, endurance, and skill. But I would just allow the target to spend a HD to heal HD + (Warlord's Cha bonus). At higher levels maybe you could let them spend 2 or 3 HD at once. Having a warlord in the party inspires you to draw on your deep reserves, letting you fight longer and harder, but you're still damn tired at the end.
Call of Restoration: I really like this one. You can try to get a PC back in the game, but the Warlord had better have a healing kit handy to stabilize them if they start bleeding out instead.

I agree that the hearing restriction is not needed. This sounds more like emergency battlefield care: "Die on your own time, soldier, we need you now!"
Tactical Leader, Bonus Action: Messing with the action economy could be the warlord's core ability, but I think it needs to be more restricted and clearly defined. The extent of these abilities also depends on whether the warlord gets Extra Attack at L5 or L6. I would be conservative with these at first. For example, you could simply give them a power like Commander's Strike X times/short rest, adding Int bonus to the ally's damage. Then you could gradually expand the uses of the power to things like: "If the target has not used their bonus action, and an ally of the target is adjacent to their opponent, they can use a bonus action to take the Disengage or Dash action". "If the target has not used their reaction, they can use their reaction to Parry". I'm just spitballing here but I think it is easier to open up new uses of this power one at a time.
We could also make the argument that this sort of thing is redundant with Battlemaster maneuvers generally, and the Warlord would be better served by having a slightly different menu of maneuvers to simulate this kind of helping.
Action Inspiration: I personally think that 5E makes it too easy to get bonuses and advantage on skill rolls; it encourages DC inflation. So I don't like this one.
Strategic Leader: Very good once the mass combat rules stabilize. It would be fun to see PCs spontaneously organizing battles when the numbers are overwhelming, and transition seamlessly from individual combat into mass combat and back again knowing that the warlord gives them a big advantage in the mass combat setting.
That's it for the powers you suggested. There's nothing wrong with going this route. Personally, the more I think about it the more I see the warlord as a battlemaster/bard hybrid. It shouldn't be a fighter subclass, because the warlord doesn't need to go up to 3 and 4 attacks per action like the fighter. And it shouldn't be a bard subclass, because it shouldn't have spells. But a good base warlord would be like the Fighter, minus action surge, second wind, and the 3rd/4th attack at high levels, then given maneuvers and inspiration dice to make up for it. On top of that chassis, there is certainly room to add in some of the powers and abilities that El Mahdi suggested. And I would certainly give it some powers to heal allies in the form of temp HP or letting people spend their own HD in combat.
Cheers,
Ben