I am using class-agnostic to mean that it is not unique to a class and is therefore open to more classes. Most class abilities are not shared between classes because they define those classes, and are therefore not class agnostic. Second Wind is not class agnostic. Lay on Hands is not class agnostic. Bardic Inspiration is not class agnostic.
If I just look at Fighter, there is Second Wind, Action Surge, Weapon Expert/Adept, Indomitable, Unconquerable. The Champion shares none of its abilities with any other class or subclass (minus Additional Fighting Style).
A Warlord class would be the same way. There would be no way for a single-class bard to opt into the Lazylord abilities if it werepart of a Warlord class.
A Fighter and a Ranger can both be a "Captain" in the lore of the game. A "Warlord" should be treated the same. It shouldn't be its own unique thing any more than a "Captain" is.
A single class Barbarian Warlord should be able to exist alongside a single class Military Officer Fighter "Warlord".