No.
It just manifests in a different way.
Ok, that's interesting, and thank you for not pre-emptively invoking the shield of opinion...
In 0-1-2-3e and to some extent 5e what's magic vs what's mundane is almost always fairly obvious. Anyone in the game can see or read or play through something and (with rare exceptions) say about any given element "that's magic" or "that's not magic".
I can think of a couple grey areas, mostly around what's affected by anti-magic: psionics - magic or not? and which spell-effects go up against magic resistance. But, yeah, generally, it's not hard to infer. Spells are obviously explicitly magical, as are, well, 'magic items.' Some Monk might be a grey area, too, a bard's singing, etc...
4e very much blurred this distinction by adding and codifying so-called "supernatural" powers for martials, monsters, etc.
4e included keywords that very explicitly called out powers as having a specific source. Most of them - Arcane, Divine, Primal, Psionic, Shadow, Elemental - were explicitly supernatural. One, Martial, was not.
When it came to monsters, though, source keywords were often omitted. So you may have a point there, if that's what you're going to focus on...
These powers, to some including me, appear to be simply magical (as in non-mundane) abilities put under a different label
Whether you take a power labeled 'Martial' as such, and are upset that it rival a magical power of the same type/level/recharge-rate, or discount the martial label because the power 'appears magical to you,' you're objecting on the lack of a clear gap between the mundane and the magical.
In the former case, the mundane rivals the magical, erasing that gap, in the latter, the mundane, itself, is erased, removing even the possibility of a gap.
meaning that magic becomes if anything more pervasive - if also more subtle - through 4e than any other version of D&D.
Which means yes, the primacy of magic is a (not the, but a) core essence of D&D across all versions.
Magic being pervasive works /against/ the Primacy of Magic, because it becomes fungible and ordinary - mundane in the sense of everyday experience. While 4e class powers didn't do that (unless source keywords are willfully misinterpreted or ignored), 4e magic items /did/, contributing to the pervasive feeling that "magic wasn't really magical" and undercutting the Primacy of Magic.