Let's break down this requirement list:
1. To heal is to restore hit points. To restore hit points is to limit the risk of a character dying. Curing is reactive — only doable after the damage was done. Temp hit points are proactive — providing a buffer so that no significant damage gets through. Healing can be immediate (cure spells), or heal over time (regeneration), recovered via rests (hit dice), or gained as a bonus effect from other actions (vampiric knife).
Curing is the simplest form of solution to the problem behind the requirement, but the requirement itself is, Keep people active and capable of fighting over the course of multiple combats.
2. The extensive amount of attacks is a means of generating damage in a fight, the ultimate purpose of which is to complete the fight victoriously. That usually means killing the opponents. Increasing damage output reduces the time to complete a fight, and thus reduces resources used (including healing). Granting extra attacks is compensation for not doing much damage yourself. You're delegating the damage to other people.
So the requirement here is: Provide a means of shortening the fight commensurate with what a typical class could do, but without requiring me to actually be the one doing damage.
3. Buffing allies covers a massive range of possible things, but for the Warlord, from what I read of it on the wiki, this largely relates to boosting attack, AC, damage, or movement. Just miscellaneous ways to increase damage output or reduce damage taken, which is a superset of requirements 1 and 2. This excludes buffs such as flight, teleportation, invisibility, etc.
4. The implied/assumed final requirement: Do so without using magic.
To add a little more thought to each of these.
1. It seems when people say a warlord should be able to "heal", they use it place of "restore hit points", which is the easiest portion of the healer's role for others to fulfill. I have nothing against a warlord healing (within a few parameters; such as inspirational healing not working in a silence spell) but to me, this should be the secondary function of the warlord, not the primary. That is at odds with his "leader" moniker in 4e, but since I don't think the warlord should be a cleric replacement, I'm fine with his healing being more on par with a paladin (more or less) than a cleric.
2. I think its important to differentiate between "granting an additional attack" and "granting an additional attack action". What the warlord should be doing is "loaning" his attack to a more advantageous ally rather than letting the ally attack again. The reason is simple; there are plenty of other classes who can get more bang for their buck than a simple weapon attack from a (hypothetical) warlord. Imagine for a minute a warlord gets no self-buffing to his attacks (no riders or extra damage dice). If that is true, than lots of abilities get better by having the warlord forgo his attack action to give it to a fighter (extra attack), rogue (SA), stunning fist (monk), paladin (smite), barbarian (rage) or even a cleric (divine strike). Thus, its imperative that the warlord NOT grant an extra attack/attack action, but a special new ability that doesn't allow classes mentioned above to trigger their special attack riders. It would be a bit metagamey (akin to how an unarmed strike isn't a weapon, but it is a weapon attack) but something like "You grant your ally an opening, he can roll an attack roll with your proficiency bonus plus your Intelligence modifier. If he hits, he does his weapons normal damage die plus your intelligence modifier." (rough draft).
3. This is the area that is hardest to define. What is the warlord able to buff? Attacks and damage, AC and saves, skills and ability checks seem fairly safe, but beyond influencing die rolls, what does a warlord do for support? How does he influence the exploration and social pillars beyond boosting skill checks? A spellcaster adds so much "non-math-influencing" effects (such as divinations, water-breathing, curing poison and disease, making food, or easing travel) that a warlord has to somehow match non-magically or be the inferior choice.
4. Which limits what the warlord can accomplish, as he would have to be balanced against some form of spellcaster (even Mearls' version was balanced against the EK) but have to create effects that are useful and don't feel "supernatural". For a fighter subclass, that is easy (since you only need a few support powers to fill in the gaps of the fighter) but the least satisfying. For a paladin-like class, its a little harder (as a paladin eventually tops out at 5th level magical effects) and for a cleric-like one, nearly impossible.
5. I'd like to add a 5th point: Warlords ultimately are suffering from the fact that in 4e, they could replace casters because caster magic generally ended up being attacks anyway (save for utility powers and rituals). Imagine, for a moment, that rather than replicate the "martial leader" role of 4e, we start with the marshal (a underpowered but interesting take on a commander class) and then layer on some warlord inspired ideas (such as inspirational healing, granting attacks, or setting up strategic maneuvers. The marshal also had a magical cousin in the 3.5 PHB2: the dragon shaman. He was an aura-based buffing character who didn't have spells but could heal (a souped up lay on hands) and gained dragon powers (including a breath weapon) and was far better balanced than the marshal. I think these two classes would make an interesting basis for a warlord, especially when you start layering on some warlord concepts to them. The PHB2 also had the knight class, which also is a good place to look for "inspirational warrior" ideas.
I think that once you free yourself form the notion that the warlord must be a low/no magic cleric, you find there is plenty of design space and ideas to work with.