But supernatural doesn’t have to be magical, particularly in a setting where magic is defined so specifically as manipulating an energy field called the weave by means of somatic, material, and verbal components.
In the jargon that makes most sense to me, magic is magic is magic is magic.
Anything that is "extraordinary" in our reallife counts as "magic", including the weird stuff like quantum observer effect and any sufficiently advanced technology.
That said, magic doesnt necessarily mean "spell casting". Sometimes magic is class features and so on.
How exactly would you give them narrative control?
The Fighter class needs "magic" by whatever name, especially to function at the superheroic higher tiers.
Ultimately it is the Fighter fans who need to decide how to do this.
Maybe each Fighter subclass defines the method of magic that continues to develop as one advances thru the higher tiers.
The choice of Eldritch Knight comes with Arcane "science" narratives, and Psi War with Psionic "soul" narratives. The default for the Fighter class and the Champion subclass should probably be the ki, the part of the soul relating to the bodily aura and body magic, similar to the Monk. Personally, am less fond of part-casters but this is a mechanical concern about it feeling subpar at the higher tiers, but otherwise the flavor of a gish is great.
Maybe the simplest way to access high level magical effects is for the Fighter class to create magic items as a class feature. These have a Zen swordsmith flavor of imbuing ones own soul into the item. Thus there is a personal bond with the item to summon it from any where, while other creatures are unable to attune the item. These magic items can have whatever effects the player chooses, and would normally be a magic sword or shield with special properties.
The ki-default Champion class would be minimalist magic, simply becoming "stronger". But even this approach might "teleport" and "planeshift" by identifying access points into the Ethereal Plane, including Fey and Shadow, whence into the Celestial and Fiend, whence the rest of the Astral.
Other methods of narrative control are also possible. For high tiers, maybe lean into the "Bastion" Fortress. Here the Fighter class can assemble a literal army of NPCs, and in this way instruct the NPCs to perform various narrative tasks.
The Battlemaster can full-on dabble in magic, simply by selecting special Maneuvers that come with magical flavor and extraordinary effects.
Ultimately it is the Fighter fans who need to agree with each other. But there are numerous ways to amp up the magic while still maintaining the feel of a traditional D&D Fighter − except of course that this Fighter refuses to retire at level 8.