I personal would be ok with "everything in the world is magic".
It's hard to argue with that one. Action Surge? Magic. Inspiring Word? Magic. Expertise? Magic. Proficiency with leather-working tools? Magic. Turnip Farming? Magic. Sun coming up in the morning? Magic. Coffee so you can face the sun in the morning?
Definitely Magic.
where I can agree with bless and charm (although the idea that magic is where 5e hides the powers annoys me that's a different issue) but dragon fear?!?!?!?
damn I hate typing that all out... no where is that said to be magic... I took it to be the rather rational response to a giant magic beast, but no where is it said to be magic that I see...
There are a lot of giant beasts in D&D, not all of them have fear auras.
I starting to miss 3.x, where things got nice, clear (EX) and (SU) labels.
is there anyone left who is going to poo poo this idea?
It's shaded more towards magical than the studiedly-neutral language of the OP, so, sorry, yeah.

how about supernatural, or extraordinary (I would prefer the second but be open to the first) instead of magic like, other possibilities come form the name of every comic from mavel... Amazing power, Astonishing power, Uncanny power, All new all different power, but I think those first two are best...
And, here, again, 3.5's 'Extraordinary' (EX) would work very well. It's explicitly non-magical but also explicitly beyond the mundane.
And, really, I don't see the problem with explicitly both non-magical & non-mundane. It can still be re-fluffed as subtlety-supernatural in a way that doesn't change it's mechanics as a non-magical ability (still working in an anti-magic shell for example - not that I'm concerned it's going to come up, it's just an example).
Also not crazy about lifting the whole bard thing on 'words are magic' or music is magic - Bards, fine, they go on and cast spells that way. Warlords, probably no singing, definitely no spells. I'd like to see more of the fluff-credit being given to the allies, too. Warlords can exhort greatness from their allies, but it's the allies that come through with that greatness.
There is a certain degree of ambiguity written into these class descriptions that intentionally empower players and GMs to determine this for themselves.
The barbarian description is prettymuch just Totem vs Berserker, the paladin's seems to imply a divine conceptual force of justice that gods and paladins are both linked to or something but if there's wiggle room for an Oath of Fealty or something so much the better, the rogue's 'almost supernatural' is still not supernatural, and that's just fine.
So where does the warlord's power come from? An arcane power of words? An almost divine faith in their comrades and martial conviction? Or is it an extraordinary ability that come from within themselves as simply the power of their own charisma?
I could see a side-bar with the always convenient 'some say...' sort of language speculating about any or every supernatural rationalization for warlord abilities. As long as the mechanics support the concept, which is martial & heroic (even if the 'Warlord' himself might not always be the hero of the story), not magical or mundane, and the fluff & rationalizations don't undermine that.