Just to touch on one point, Hit Dice are effectively healing surges. My warlord I'm working allows others to use their hit dice (with a bonus) with Rallying Cry.
I'm not a fan of the warlord triggering them for a couple reasons.
First, because it doesn't work well with alternate hit dice rules. For DMs that want slower healing the warlord becomes problematic.
Second, and more importantly, it doesn't actually extend the adventuring day. It serves as some emergency healing, but the party can't adventure more than they could without the warlord, having the same healing per day.
Third, it does quirky things to the resource management of the game. It means one player can't use one of their class features if another character poorly managed their resources (i.e. hit dice). And it means players are less incentivised to take a short rest, since the warlord can heal otherwise, which might penalize character that recharge powers on a short rest, as fewer party members will benefit from the rest.
No other class feature touches hit dice. No feats affect hit dice. Most options related to that mechanic deliberately refer to the vaguer "regain hit points during a rest", because how hit dice function are left out of the assumptions of how other mechanics works. This is part of the modular design of the game.
Now, working based on a particular assumption of the rules is fine for your home game. But it's poor when designing something to be used at any other table. It'd be like designing a class that assumes the existence of magic item shops, or the use of the honour statistic, or point buy vs rolling for stats.
Personally, I'd love to design some monsters assuming the Sanity ability score was the norm. But that'd just limit their usefulness.
I disagree with anyone being able to do it. Martial abilities can and should still be unique to classes. Why can't the bard just take another action like the fighter's action surge? The inspiring leader feat assumes something more than just a high charisma to grant temp hit points. Martial doesn't mean anyone can do it if they roll high enough.
If we're working under the assumption you can talk someone into regain hp, I don't see why should that be limited to the warlord. It's not something that requires physical mastery or skill that the uninitiated cannot even attempt, like a sword move or acrobatic trick. What can the warlord possibly say that the bard or the paladin cannot? What if the warlord told the bard what to say? Would it work then? Is it really so impossible that the bard could not attempt it, roll a natural 20, and do just as well as the warlord?
I agree that martial abilities should be unique to those classes. But there's a big difference between the specialized action no one can attempt (action surge, sneak attack), the things martial characters should be better at but anyone can try (multiple attacks, shoving), and the stuff non-martials can do almost as well (making a melee attack).
Inspiring someone with a speech doesn't sound specific enough to be limited to one class. It's almost something anyone can attempt, be they wizard or rogue or even barbarian. Especially since inspiring speeches in fiction can be given by anyone given the opportunity. Yes, the warlord could arguably be better at it, but that doesn't necessarily mean everyone else should be denied the opportunity to try.
If you're going to say that you can heal through morale, go all in and give it to everyone. Especially the bard who literally has a class feature designed around inspiring people through talking. (Arguably, the bard was also healing through inspirational words in 4e. )
The warlord and the bard overlap a lot. Really, so much of what the warlord was in the 4e PHB1 worked solely because the bard wasn't out yet. It filled the niche of the charismatic leader. (And necessitated the bard being more magical.) But with a bard already in the game, and a bard able to smack people with a sword and rely on strength, the charismatic warlord seems less necessary. That archetype is filled. Having a Charisma focused warlord and the bard is like having a scout and ranger. The two classes need to be pushed farther apart and the unique elements of the warlord brought to the forefront. And what makes the warlord unique is not its healing. It's everything else.
The warlord seems better served focusing on the tactical leader. The strategist. The intelligent fighter. Charisma is nice, but seems less necessary. Being charming isn't what makes a warlord a good leader, and doesn't work as well with the tactical features that are unique to the class.
That's the difference between making a good interpretation of the strategist archetype and making a good update of the 4e warlord class. I'd rather see the former.