But why? Why cannot non-magical character have daily powers? We already accept them having encounter level powers - Second Wind, Action Surge, etc.
Casters, by double digit levels, have a fairly lengthy shopping list of encounter ending powers. I hope we can agree there. So, why can't a fighter EVER just kill something? Why can't a high level fighter have a 1/day Death Stroke attack that instantly kills something if it fails its saving throw? We have zero problems with every caster being able to do that. So, it's not a balance issue. But the notion that your fighter can just instantly kill something, or intimidate stuff into running away a la a Fear spell, x times per day? Absolutely not. Must never, ever happen.
A fighter can do a ton of damage
against which there is no save multiple times in every turn just by attacking with their weapon. Often, this is enough to kill an opponent. In terms of average damage/round, which is the most efficient way to kill enemies, few if any classes can match the fighter - paladins and barbarians are likely the runners up. I've been running D&D campaigns for decades, and in my experience martial classes have far and away more killing blows than casters - the only time casters do better is if it is against a ton of weak opponents and the caster has suitable AoE prepared.
So I'm not even sure what you are comparing fighters to, here. What is the comparable death-dealing power that "every caster" has? What spell do they have, castable round after round, that does comparable damage and kills? And what does "high level" mean?
Like I've repeatedly said, this argument was lost years ago. It was lost because of counter arguments like yours. Non-casters must never be allowed to even come close to what casters do. I actually tracked it in a couple of different campaigns. The casters were doubling or even tripling the total damage output of the fighter types over time. A fireball doesn't deal 25 points of damage. It deals 25 points of damage to five targets. Something even a high level fighter would struggle to do and is flat out impossible for any fighter with less than 5 attacks.
Fireball is an exceptionally good spell against multiple opponents, probably too much so. Even so, it's terrible against a single opponent. So let's say the opponent is the BBEG - kind of an important fight! Suddenly your fighter and other martial types are going to be doing most of the damage. Where is the caster that can maintain damage output like a fighter can, round after round, all day long if needed?
Fighter types, under 10th level, are dealing about 25 points of damage/round (give or take). Casters? Generally batting about 40 overall. And that's just dealing straight damage. That's not even counting stuff like Banishment or anything like that.
Really? Your wizard, bard, sorcerer, cleric, warlock, or druid is averaging 40 damage per round at levels 1-10? Round after round? Against any number of enemies? Because the fighter types are maintaining their damage all day long against any number of foes.
DPR is not a good metric to base this argument around, because martial classes win that comparison hands down, fighters in particular. They are the DPR S-tier class. Martial classes lose out on flexibility, not damage dealing.
Put it this way. Which group would be more capable, a 15th level group of 4 PC's with two fighters and two rogues (all without spells from subclass) or a 15th level group of 2 clerics and 2 wizards? It's not even close.
What are their sub-classes? What are they fighting? Why aren't they allowed to have spells from their sub-classes? Are they fighting a dragon? Fighters and rogues!
But if you were asking me to build an optimal party of 4, with the most capability for handling a variety of foes, it would not be either of those combinations! It would be a fighter or paladin, a cleric, a wizard or sorcerer, and...hmmm...rogue/bard/or ranger, maybe. Though a recent campaign had a cleric, wizard, fighter, and barbarian, and that double tank combo was a huge pain in my butt to deal with, especially with cleric support.