Edit (2) : Even the warlord in 4e wasn't just a buff/debuffer... he was able to competently attack & do damage...
Not that competently. A warlord, by himself, was pretty weak. Basically he only had cantrip level damage.
Sure, some people played the warlord like a battlemaster, focusing dealing damage, toughness, with a side of buffs. But it's much more that he was buffing with a side of damage.
They are closer to a valor bard then a fighter.
And the lazylord is similar to a pacifist cleric. All the images of a cleric show them waving a mace around, or burning people with fire. It has all sorts of offensive spells. Yet some people choose to avoid them and only take defensive spells. Eventually support came along for that kind of cleric, and it got it's own class in 3e (healer). You can readily make a pacifist life cleric in 5e, with only 1 small feature being "wasted".
Edit: And to answer your EK question... the biggest thing is free access to spells outside the evocation school...
They can take magic initiate and get any spell. Which is the same type of argument you give.
Eeven if you could take other spells, it would still be primarily a fighter, with only some wizard.
I'm asking what actual abilities in a properly built/spec'd warlord are missing that are necessary for him to be a warlord... and you're giving me vagueness. So I'll ask again, what can't the Battlemaster do that the 5e warlord should be able to do?
Would should a fighter do that a war cleric cannot do?
He has heavy armor, martial weapons, and a few times per day he can multi-attack.
The answer, again, is that it's still primarily cleric with only some fighter.