Not to get too deep into the weeds, because I've never actually played a 'Warlord' using the Bard chassis... but it all comes down to what I would find most important.
I kinda did, actually, back in 3.5 - built, not played, because the build ultimately was not satisfactory - but the concept wasn't warlord. It was, humorously, "back-seat adventurer" that is, a Sage who knew all about monsters, treasures, history, tactics, etc, on an academic level, but no experience or talent for the actual work of exploration and little for fighting. The initial thought for a Sage was, of course, Wizard, but that came with so much spell power it'd be the inevitable focus of the character. So, Bard, it's 'song' giving everyone a bonus could be re-skinned as annoying advice - all the more annoying because it sometimes helped! But, ultimately, anemic a full caster as it was, the 3.5 Bard was still a full caster, and it wasn't practical in terms of either available choices or overall contribution, to downplay that enough to fit the concept.
An actual warlord would run into the same issues. Re-skinning spells to the point they're immune to anti-magic &c is squirrelly, and downplaying them enough to fly under the not-magic-I-swear radar would leave you under-contributing.
Heh, yeah, except of course the irony being that you'd get three Warlords that were much more mechanically distinct by using a refluffed Bard, a refluffed Cleric, and a Fighter Battlemaster as your Warlord options than you would by having WotC make a single Warlord class with three subclasses, LOL.
Mechanically distinct in the sense that the BM would be profoundly inferior and unable to pull it's weight in a support role, while rivaling the dedicated melee types.
OK, and in the sense that the two superior warlords are shut down by antimagic.
And the annoying way you'd have to willfully pick sub-optimal spells every day as the cleric (at least the Bard need only do so at level-up).
Ultimately, the Bard is a fair touchstone for how powerful/effective/versatile a warlord class would need to be for basic viability, tho.
But you also have to remember who I am here-- I'm the yo-yo here on the boards that doesn't really give two craps about game mechanics... what matters to me is the story of the party and the adventure. So I have absolutely zero concerns about re-fluffing, and any re-fluffing that doesn't match up 100% from the mechanic to the story I will happily handwave away. Because once the mechanic has been used... it's the resultant story that came out of the mechanic's use that actually has import and is what we care about and remember.
And this is absolutely something that most other people don't do, don't agree with, and argue with me about.
Re-skinning is awesome, yes. 3e let you do some explicitly (you could describe your character & gear as you liked, re-skinning spells apparently required Spell Thematics - better than 2e, where you needed to cast a spell, Sense Shifting, to re-skin other spells); 4e you could re-skin more (even spells) and very easily, since fluff was separated from crunch to a degree, but you couldn't change keywords which means re-fluffing Source was out; 5e

ask your DM the rules are vague with little to differentiate re-skinnable fuff from sacrosanct rules text (and, even then, the rules text is often vague and begs for rulings).