Find me one warlord in any book whose battle field role is "healing"
IMO 4e invented this and it's been nothing but headaches since. I played one and loved it, mechanically, but at that point I didn't care because I'd long since given up trying to play 4e as anything but a turn-based combat game that has very disassociated mechanics to story. My DM didn't even grok the warden's marking ability, and that was supposedly magical! 4e made warlords heal better than a magical cleric...what? Warlords rocked in 4e, so much so that they completely crushed clerics out of the picture. That is a HUGE D&D design fail right there. Warlords should be good at helping you win the battle, not patch up their comrades. A warlord should "heal" stuff like mind-altering impediments, lifting the fog cloud of some evil wizard or curse by reminding them what they stand for and so on. Or, if they do heal, that healing should be strictly limited and require you to be literally next to them. I actually prefer some of the reaction-based prevention of damage, which is a mechanic that ends up doing the same thing if not better (an ounce of prevention is worth a pound of cure!). He should be able to shout out "duck, man!" or "do defense drill #5", and allow you to boost your AC for a given attack.
When people say they want warlords to heal better than clerics, I shrug and think to myself, keep on playing 4e then. Or if you want a martial class to pick up healing ability, do it strictly via the healer specialty, case closed, there's your warlord + healer. Not all warlords should have healing, especially if they don't want / need to. The real problem is more that clerics were being overshadowed in the "leader" role of 4e, which lumped in all sorts of things. You want a support role class that does everything? Be a cleric. You want a support class that focuses on combat offense/defense-oriented tactical boons? Play a warlord. The problem here is that 4e pre-defined "roles" are creeping back in, and that leads people to think warlords should have functional true healing at range despite having no magical "spooky action at a distance" hand waving explanation. When you have to conjure up a contorted mechanical explanation for non-magical healing, especially during combat, that's where players of all editions except 4e rolled their eyes. And I LOVE warlords, probably because I'm a power gamer. I didn't see a reason to even play a full on warlord, I could do much better as a ranger|warlord hybrid and use my reactions or minor actions for warlord powers. A dedicated warlord build was uber powerful in 4e, which is right there ipso facto the reason they were played so much, not because clerics are "played out" or a boring class to play per se. At least, they shouldn't be a boring class to play. And if I can pick up a very good healing kit as a martial class, why can't a rogue be just as good a healer as a warlord + specialty?
IMO magical healing HAS to be better than non-magical healing. Period. Anything more than that requires too much unbelievability. Especially instantaneous healing at a distance. I really don't see what you guys' problem is with warlords preventing party damage before it happens rather than repairing it after it does. If you design the game that way, then it actually makes sense to have both a warlord AND a cleric in the party, and avoid the "too much healing, not enough damage" problem you'd get, since the warlord can focus more on damage-boosting boons than defense ones when the cleric is around. Yes, I get the imagery of "healing the spirits" through a good chat, but modern science shows that all that fancy good-will doesn't actually change e.g. cancer death rates one iota. Let's let magic handle the "magical" stuff, and leave non-magical stuff work the way you'd expect it to in the real world. In real life, if you apply bandages as a field medic, that only prevents the damage you've already taken from getting worse (stop the bleeding, put a splint in).
What the 4e warlord presupposes is an entire 21st-century ER contained within the charm of his sweet, soothing words, that can take you from 0 HP to full -- instantaneously, and at a distance!! Not even Star Trek technology can do that!!!
i.e. == IT is complete and utter BOLLOCKS