I guess the more I look at it, the more I wonder why "Warlord" isn't just a War/Life Cleric 1/Valor Bard X, Fighter/Valor Bard X, or Battlemaster 3/Valor Bard X.
OK, think about a Valor Bard X/Cleric 1/Fighter 1. You get:
bless
guidance
resistance
Combat Inspiration
healing word
heavy armor
martial weapons
Song of Rest
Non-spell healing (with Life Cleric 2)
Protection Style
You don't need high Wis and you don't need very high Cha, so you can take higher Dex, Con, and Str.
Be a Half-Elf. If your DM lets you use SCAG variants, be a half moon/sun elf and take the Elf Cantrip for booming blade for extra battlefield control.
Take stats:
Str 16 (15 + 1)
Dex 10
Con 13
Int 8
Wis 13 (12 + 1)
Cha 16 (14 + 2)
Start with Bard. Gives you the most overall, although you can take Fighter first if you'd rather.
At level 2, select Life or War Cleric. Gain Medium Armor, Heavy Armor and Shields. Gain healing word, bless, resistance, and guidance.
At level 3, select Fighter. Gain Protection Style. Gain simple and martial weapons.
From here on, take Bard. Select College of Valor to get Combat Inspiration.
At level 4, ASI to increase Con and Str, or select Inspiring Leader or Shield Master.
At level 8, you get Extra Attack. Late, but booming blade should have made it not too painful.
Now, yes, I can easily see arguments for dropping Cleric or Fighter, depending on the style you want. It's up to you. This is kind of the all-inclusive build. You gain a fair bit by taking Fighter 2, Battlemaster 3, and Life Cleric 2, too, but Bard is the bread-and-butter. War Magic is a bit too deep into Eldritch Knight to be worthwhile, however.
No, you don't get all these abilities on reactions. Essentially nobody gets good reaction abilities anymore. There's Riposte Maneuver (Battlemaster), Misty Escape (Archfey), Defensive Duelist, and that's basically it. No, you're not handing out Action Surges. I mean, let's be honest, if we ever get a 5e Warlord, he's not going to be the Action Economy Trading Post that he was in 4e. If 5e ever has a 4e Warlord-style "grant an action" ability, it will -- at best -- allow: a) a single attack, b) a cantrip, c) some amount of movement. If it happens off-turn it will probably take the target's reaction to do so, and might still be limited by the Charisma mod of the Warlord. The Bard's ability to grant bonus dice is a reasonable approximation. No, d6 or d8 isn't as big as an extra 1d8 + 4, but numbers in 5e are a lot smaller overall.
Now, if your whole argument is, "But I don't want to do it with spells!" well, I'm sorry. 5e reverts to the old D&D model and puts most special abilities into the "spell" column. That's why 4e made some people feel like every class was casting spells.