I have to admit, I never considered warlords as a kind of fighter.
For a long time, the only 'Martial' character was the fighter. So it's easy to see 4e as having taken the fighter and 'split it up' into the Martial Source classes - with the exception of the Rogue, of course:
The fighter kept the traditional toughness and meat-shield duties, but with teeth, actual class features that gave it 'aggro' the way everyone was clamoring for throughout the epidemic of 'fighter SUX' threads on Gleemax, plus greatly expanded versatility (at the price of customizeability relative to 3.x) and more resource-management & agency in general.
The Ranger took the traditional 1e/2e Cuisinart-of-Doom TWFing build and the often-marginal archer and made them viable (but not too broken) DPR machines.
The Warlord walked off with just the classic fighter's name-level ribbon, and expanded it into a class that finally supported so many of the archetypes the fighter had always fallen so far short of (and a few more the designers never even expected).
About the only thing lost to 4e's kerf allowance was the 3.x 'battlefield control'/'tactical reach' builds, which would've likely violated the sanctity of the wizard's controller role too much.
It engendered the opposite sort of confusion, too. People were like "Fighters 'can't' use bows? WTF?" when the Ranger was that edition's non-magical archer, for instance, because the ranger's Martial status was easy to miss if you were expecting it to be casting spells as in all prior editions.
So, yeah, there's a strong bond between the martial source and the fighter, and it's created tangled expectations about what names should be used where and how the design space should be laid out.
But, 5e very clearly laid out the fighter's design space. It de-facto tanks (for want of any 'aggro' mechanics, but traditionally many DMs just have more bad-ass enemies attack the fighter for honor/glory/pick'n-on-someone-your-own-size/whatever) and solidly delivers DPR, and has precious little room for anything more. And, one of the three things they put into that precious little space was Casting Wizard Spells.
So, the fighter class can't be used as an analogue for 4e's Martial Source, instead, characters that would have simply been Martial in 4e must be defined by their lack of supernatural abilities.