The fact that various editions of D&D have explicit mechanical options to do specific magic things does not seem a strong argument for characters without such explicit mechanics to also do those magic things.
Well, the question about fighter's fireballing (something they can do) was a gotcha, whataboutism anyways, however, you seem to be missing the point.
If a Battlemaster wanted to cast Fireball exactly like a wizard, and had no spell slots and no fireball spell known through some special ability, then I would say no. Because unlike Divine Magic where an entity in the story world would reach down and perform the miracle, Arcane magic needs to be taught and learned and studied.
However, there was a reasoning in my actually semi-serious answer. I'd let the Battlemaster with Arcana spend an hour long ritual, with a DC 15 or so Arcana check to spend multiple Hit Dice to cast fireball.
Because what is the narrative difference between a Fighter who at level 3 becomes an Eldritch Knight with Spell Slots because they have been studying magic and got good at it, and a Fighter who at level 3 becomes a Battle Master without spell slots because they have been studying magic and were not good at it? Arcane magic is explicitly a skill. Wizards are not born, they study.
So, the question isn't "is it possible for someone to cast fireball with just the knowledge of the spell formula" because that seems to be obvious, of course they can. The question is, what are the consequences and how long does it take?
Now, if they weren't proficient in Arcana? Then I'd tell them there was zero chance, just like if you handed me some Python code, Unreal Engine 3 and said to make a video game I'd tell you there is zero chance. That's all gibberish to me, I'm not a coder.
It isn't about resources when it comes to fighters creating spell effects, it is about narrative positioning. Amusingly, Primal Magic is the one I see least likely to work. Elemental and Bestial Spirits are far less likely than gods to be forgiving to someone stumbling over a ritual, and Arcane magic is explicitly just a honed skill.