First, there's the concept of the warlord as the tactical leader, which doesn't mesh well with the concept of the field medic. They're two very different ideas that don't mesh well together
That's the result of trying to appease the spurious objections to hp-restoration mechanics modeling Inspiration. Inspiration, tactics, & leadership all go together neatly. "Field Medic," conceived as a less abstract way of restoring hps, more literal 'healing' or wound-treating, is irrelevant to the class concept. That's what you get when you try to take input on class design from people who hate the class on a conceptual level: incoherence.
Adding healer to the "commander" class necessitates the removal of class features for balance.
Not necessarily, it depends on how abilities are chosen and used. It's like adding a spell to a class list, it slightly increases flexibility, you don't usually have to take away another spell to compensate.
Second, in-combat healing is a bit of a waste. It's inefficient as you can almost never do as much healing to the party as an enemy can dish out in multiple rounds.
The most critical aspect of in-combat healing is getting a fallen ally back into the fight. Numbers tell very heavily under Bounded Accuracy, and a PC victory usually hinges on taking enemies out of the fight, thereby shifting those numbers in their favor. A party that can't bring an ally back into the fight is very vulnerable to the reverse.
You're better off removing an enemy from play and taking future damage out of the equation.
The most efficient healer in the game is a striker.
That's one valid play style. There are others. If that's your playstyle, you'd want to make choices to emphasize offensive buffs. You'll likely have the odd TPK, but that's just part of the sense of challenge that makes that style fun for those who prefer it.
Third is that the cleric is not automatically a healbot.
Traditionally it was, and with the healing burden spread out over multiple classes in 5e, depending on who else is in the party, it can be a lot less of one, even ignore healing entirely. The same would be true of a Warlord. It would need to have the ability to provide adequate support to the party if it's the only such class there, it should also have the flexibility to shift that focus should there be one or more other support characters (other classes or other Warlords) in the party.
Only one of the six subclasses of cleric assumes healing. You can play a cleric that doesn't heal very easily. Or one that only heals during downtime. It seems odd to make a class that's more focused on healing than a cleric
That would be both odd, and inappropriately limiting. That the Warlord must be able to restore hps in order to enable certain playstyles doesn't mean that every Warlord would have to choose, focus on, or ven use that ability at all.
This is also made muddier in that the bard and druid are also full spellcasters and have access to the same amount of healing as the druid and similar spells. So either class becomes an acceptable replacement for the cleric. So the warlord as the class for people who don't want to be a cleric is less necessary as there are two others already.
One of them a nature-priest, so barely differentiated from the Cleric as a concept, and the other a minstrel - and they're /all/ spell-casters.
Fourth, healing is not the be-all end-all of clerical abilities. 5e is different from 4e where almost no conditions persisted beyond and encounter and healing was all a leader needed to do.
Meh. 5e is not such a bad game that it can't handle the Warlord. 4e was not such a bad game that nothing from it can work in 5e. The edition war is over, 5e is D&D for everyone who ever loved D&D, and that includes 4e fans who'd like to play a Warlord. 5e is up to that challenge, lingering edition-war prejudices and nay-saying notwithstanding.
Now, lesser restoration and greater restoration are pretty essential, removing a wealth of negative status effects. Without those spells, the warlord will be less useful and cannot fully replace the cleric.
That's assuming spells are the only possible solution, which you seem to think is the case with restoring hps, as well, but the Warlord could do that. So it might not be an issue at all. But...
Neonchameleon's case is that the Warlord would enable play in low-/no-magic games where spellcasters are inappropriate. Most of the sorts of effects that restoration copes with would also go the way of spellcasters in such a game, so it's a non-issue in that instance. In a higher-magic game, a ritual caster or even NPC could cover lingering out-of combat effects. In combat, the Warlord might offer ways of mitigating or minimizing the negative effects of such things, through inspiration and/or compensating for (or even leveraging) them tactically. In an all-martial party in an otherwise standard World, that last idea might have to be depended upon until an outside resource could be found to remove the effect permanently.
There's no question the Warlord wouldn't be exactly as useful in every circumstance as each of the existing support classes (which are very nearly fungible, because they do cast many of the same spells), which is only to be expected, as it is a very different and unique concept not yet covered by the game, but that doesn't mean it can't adequately contribute support to keep a party going through the 5e adventuring day. And, it does make it's addition all the more desirable.