At-will attack granting can't be broken because of Rogues and Paladins (and Rangers?) because having a second Rogue or Paladin in the party, or even an all-Rogue or all-Paladin party, isn't broken. Now, if the action granting ability is sitting on a chassis that also gives a wide range of off-turn or out-of combat abilities, like a full caster, that could easily be broken.
Those classes do not also heal though, at least as well as a cleric. Something has to give either the attack granting or the healing rate and I do nto think attack granting at will is a good idea in 5E (you could use more superiority dice than the BM fighter) because of how it interacts with things like Rogues (any), Hunter Rangers (basic atack + 1d6+1d8 is great vs clerics anaemic damage).
A battlemaster fighter with the right 2-3 feats is probably better than anything Mearls can come up with anyway. A proper 5E Warlord needs to lose at will attack granting and be an independent class for the same reasons a 5E wizard/CoDzilla can't do everything they could do in a previous edition.
The 4E fans basically expect 5E tp make all the sacrifices to enable an OP class when other classes made sacrifices to fit into the 5E design paradigm. Even in 4E a warlord was not as good at healing as a Cleric so you can start looking at perhaps Druid or Bard levels of healing.
For example look at he 5E Paladin a warlord should heal better than that yes? So strip out lay on hands and the spells that gives you some design room, strip out the smites the extra attack and Paladin Aura and thro in perhaps X2 or X3 healing via a lay on hands replacement, more superiority dice than a BM fighter (+50-+100% more perhaps) and you still have room for support abilities.
A Warlock chassis is another class I think you could write in WL abilities instead of invocations and replace spells. A Bravura Warlord could still get a 2nd attack at level 5 or 6 jut not the extra stuff classes like Fighter/Ranger/Paladin get to make their attack better.
If you want at will attack granting at best you get Paladin levels lay on hands healing, and Battlemaster Fighter levels of anything else tactical.
Put it another ray the warlord gets 4 pillars and you can rate how strong you want the warlord to be in the 4 pillars on a scale of 1-10.
Attack Granting
Support
Healing
Combat
At will attack granting costs you 10 points, 10 points of combat gets you fighter/paladin/ranger levels of combat, 10 points of healing gets you life cleric level, 10 points of support gets you perhaps a buff focused lore bard level of ability.
You get 20 points. Knock yourselves out.