A class must go
beyond the zero point. It doesn't have to be that far. It certainly doesn't need to be at the level of Wizard/Sorc, Cleric, Druid, or Bard spells. Tactical Mind falls short, but not by a huge margin; the issues are largely its few uses and requirement that you sacrifice critical healing resources to power it. Imagine if Wizards had to sacrifice hit dice to cast spells! Wizards would be howling for blood. By comparison, the old Remarkable Athlete was dramatically worse than Tactical Mind, and the Battlemaster getting one niche tool proficiency was likewise inadequate. Reliable Talent, from the Rogue, is actually a very
good class-derived tool for skill stuff; coupled with Expertise and the ways various subclasses contribute, Rogue is mostly fine on this front.
Spitballing a Fighter feature I would consider appropriate, with the caveat that
this has not been tested, I cannot promise it would be balanced, I'm thinking of something like this:
Gritty Determination
At 3rd level, your single-minded determination to see a task completed carries you to greater heights. You have a pool of Grit points, equal to your highest ability modifier (minimum 1) plus your Fighter level. Any time you make an ability check that isn't an Initiative check, before you roll, you may spend points from your Grit pool to increase the result on a one for one basis. You cannot add more than your proficiency bonus to any single roll. You can spend Grit points even when you have a feature, such as Expertise, which allows you to add twice your proficiency bonus. You regain any spent Grit points when you complete a long rest. At 7th level, you also regain any spent Grit points when you complete a short rest.
Now, perhaps it should be limited to half your proficiency bonus, or have fewer points, or whatever else. As I said, it would need testing. But the core idea here is simple, and this is a straightforward, easy-to-use feature. You have to invest many levels into Fighter before you get the short-rest recharge, so there's little fear of a lame multiclass dip. It's generically useful; it applies in cases that nobody would get proficiency in, which gives it unique utility; it stacks with other buffs like Expertise, Bardic Inspiration,
guidance, etc., so there's no worry about checking compatibility; and you spend the points
before you roll, so there's no "oh, I rolled a 2, there's no point" issues that I know annoy some players.
I am confident
some kind of ability like this, even if subject to some balance changes, would be perfectly functional on the Fighter. It would be unique, useful, distinctive, and fairly easy to use. And this ability, while quite useful, is certainly lower-power than being an actual spellcaster.