All of which is post facto rationalization. The mechanic says what happens, so the fiction will just have to bend to accommodate no matter what. If you can't recognize that's a very different aesthetic, well, we just won't be able to have a productive discussion.
It is a very different aesthetic and mechanic but it doesn't have to apply to only spells or magical effects.
Some mechanics are modeling the outcome and NOT trying to one to one match the player resource to the character resource. In fact that's the whole point of some narrative abilities.
So in many cases "the move" is not fiction bending to your
charcter's will, but the fiction is accomodating the outcome -- which if the mechanic is good should more often than not make sense within the character's permission space, general abilities, the genre, etc. Yes, it can lead to some situations where there is a disconnect and you have to squint or just move on. It's a downside to this mechanic. But if the mechanic is good within the system, this should happen much less often than more often.
The extreme example of this are plot points / fate points that allow players to introduce authorial fictional elements like "of course my buddy from college works at this law firm" or whatever. In a real world based modern magicless Fate game there are no spells involved and the character itself is not bending the fiction -- the mechanic just allows the player to declare an outcome related to the character that is plausible within the fiction. The character experiences a plausible meeting with an old friend.
D&D narrative abilities have traditional used magic as a way to tie the outcomes directly to push button character resources which are also one to one player resources but it seems like 5e has experimented a little bit on this as well -- backgrounds that allow the player to declare the "outcome" that the peasants will shelter them, etc.
I can see why some people are ok with this and some are not.
If it involves bending fiction to your will by performing the move, or is balanced by limited access to some metagame currency (slots per day, rechargeable points) then regardless of what you color it as, it is a spell-like mechanic. It is mechanics we are talking about here.
Then call it spell-like mechanics.
If you are using "Fighter gets spells" to mean "Fighter gets abilities that use mechanics that have been traditionally reserved for spells in D&D, particularly limited use in exchange for power and narrative control" then I understand that. And I can see why some people may or may not like this.
If people say "Fighters get spells" as short hand it's misleading and confusing because we have pages in the PHB describing what spells and spellcasters are in D&D through the lens of the D&D fiction.
