It's not just that there's few choices (BM is fairly customizeable, you could do a number of quite different ones), or that the Monk doesn't count (Ki is explicitly magical in 5e), it's that they are such very limited choices in capability & contribution. No appreciable support or control, little beyond skills out of combat, and all high-DPR in combat.
And SCAG. But they're all sub-classes, and what even a gifted designer might do with a Fighter sub-class is very, very limited.
Non-magical healing is distinctly inadequate in combat - out of combat, HD are distinctly lacking, too, while, conversely, overnight healing might even be excessive (but is at least convenient - in a typical, mostly-caster party with ample magical healing, overnight healing just saves you from taking an extra full 24 hr cycle to blow most/all slots on healing, the recover them all, in the absence of magical healing, it's the only way to get back all your hps after an even slightly-challenging day, and two full 24 hr cycles are prettymuch required to get back all those HD that you'll burn through the next day).
Even in 4e there was the generally-assumed, if not absolutely necessary, Leader Role. There were just a wider range of choices to fill it than the Cleric, Druid, Bard, Paladin of 5e.
5e desperately needs a Warlord to step into the support function in low/no- magic campaigns or just for players who don't particularly want to go for the existing support classes as such. It also needs more non-magical battlefield-control, like the eponymous reach-based fighter builds of 3e could provide, though preferably not quite that laser-focused.
What it doesn't need is even more sub-classes locking up even more of the very few things a non-magic-using PC might do, in 5e, which, sadly, is exactly what we're getting. Instead of breaking CS dice and maneuvers out of the BM sub-class, and making them into a sub-system with some of the flexibility and design space that spells have been heavily leveraged to build so many classes with, or even breaking out just the CS dice in silo'd sub-class-locked-down abilities, it sounds like the cavalier and other fighter sub-classes are just going to take narrower concepts and gate them behind sub-class.