Warlords could do healing adequately. Every other leader in the game was better.
Definitely not true. No other leader touched the most extreme Cleric healing-focused builds, like the Pacifist Healer, but Inspiring Warlords could do a crazy amount of healing, well beyond the 'adequate' baseline of all leader builds.
And temp hp work just fine in combat. It soaks a hit and prevents you going down.
That's not working fine. If you don't predict who will be hit and for how much, they're a wasted resource, and when someone does go down, the can't bring him back up. That is an irredeemable failing.
Plus, it lengthens the adventuring day unlike triggering Hit Dice, as spending HD in combat is the same daily healing as spending them during a short rest.
Both temp hps and hp restoration would be reasonable features - and, it had both, so there's no reason not to. Separating day-lengthening and standing fallen allies in combat into temp hps and hp-restoration, respectively, would make a lot of sense.
A 1st level cleric or bard can cast cure wounds twice. Or something much, much more interesting. It is literally a third of their class features.
It's a fraction of their features, since every other spell has an equal claim on being a 'feature.'
The iconic parts of the warlord are {restoring hps in combat via Inspiring Word}, imparting movement, granting extra attacks, boosting initiative, bonuses to attack, damage, rerolled saves, etc.
Which of those are you going to give up at level one for healing?
None, and there's no call to do so. Druids don't give up shapeshifting or casting Call Lighting to cast Cure Wounds, now do they?
Healing was the least interesting thing a warlord can do.
But the most iconic and critical. Healing Word wasn't taken away from the Cleric.
Like I said, it'd be better as something you can opt into.
By definition, as an out-after-the-standard-game optional class, the Warlord, as a whole, will be opt-in.
There is no need to 'compromise' with those who object to the Warlord, as they already got exactly what they wanted: no Warlord, at all, in the PH.
Not quite the same effect space - the Warlord's effect space is more ranged than the Battlemaster. The Warlord is effect-wise comparable to the Valor College Bard, and the Priest of War in range and breadth of effects (albeit the Warlord in 4E is actually less broadly competent than either 5E caster).
No 4e class had the sheer versatility enjoyed by the 5e neo-Vancian casters.
Now all the powers and features that just deal with movemnt (slides, swaps, pushes) don't translate well to 5th as positioning doesn't matter as much. That's Wolf Pack Tactics, Viper's Strike, Leaf in the Wind, White Raven, Onslaught, and Pin the foe.
It's not that those things don't matter, it's that they're handled with less granularity. WPT, for instance, let the Warlord shift an ally out of reach of an enemy, enabling him to open up distance with a regular move on his turn. In 5e, this translates to letting an ally disengage, saving him an action. That certainly matters, even if shifting is no longer, strictly speaking, part of the game.