I feel that the Warlord was specifically made for 4e to take advantage of 4e's... uh, idiom. The grid, the tactical slides and pushes, the making others fight instead of you, triggering healing surges (all leaders did this, but Warlord had the most after Cleric, I think?), all of it.
Artificers had to have the most slides and pushes, followed by bards.
What warlords where first in is granting attacks and offensive bonuses. About 1/2 warlord granted movement was part of the attack (i.e. move and attack), and extremely little enemy movement. They also has a smattering of defensive bonuses.
"The most heals" (not the "best heals") is true for a specific build, which traded off of the attack/buff/movement options. And even then, it was only true near the end of 4e due to the glut of warlord abilities and a few unbalanced feats. For most of 4e, and most warlords, it was simply surge + 1d6.
Basically, a warlord use various forms of haste. (double speed, +2 AC, advantage on Dex, extra attack), and can let someone spend a hit dice with +proficiency bonus each short rest.
5e is less tactically focused, and YMMV, but a guy built for battlefield pushes and slides doesn't seem at-home in this edition. I actually kind of resent the attempt to wedge in 4e's grid-based paradigm to 5e's otherwise free-form style, but I get that WotC wanted to please as many people as possible, too.
Grid doesn't have anything to do with it. The biggest change 4e made was the number of rounds combat lasted. In 5e, there's much less reason to slide a kobold to a cliff to be pushed off if you can simply stab it. In 4e kobolds could survive long enough to make it worth while.
The need of a rogue to care about positioning, doesn't really undercut the tactic of the barbarian running in and hitting stuff. Similarly, a warlord letting the party run about without OA's won't stop a champion archer from standing still and filling everything full of arrows.
