First, applause for making this a "+" thread so it (hopefully) won't get bogged down in the same old debates.
The short answer: get rid of Vancian spellcasting.
The long answer: the issue between magical and non-magical classes is, at it's core, based on a perceived difference in resource allocation. The classic "martial" doesn't have many abilities that are per diem; what resources they expend (other than the ur-resource, hit points) are generally far more easily regained than spell slots.
Because spell slots are a limited resource, design follows that they have a large impact when used; the Fighter does more damage consistently each turn all day long than a Wizard's fireball, but the fireball has a huge splashy impact on the battlefield to make up for it.
(Obviously, some classes skirt the line here; the Barbarian is based around a powerful daily resource, and the Warlock is based around getting spell slots back easily.)
So the first step in this solution is to get everyone on the same resource track. If everyone is balanced against doing their "thing" each turn/encounter, there's not going to be an issue where a spellcaster can use more resources than intended, ie, a low or no-encounter day, where they can transform their excess spell slots into out of combat advantages and increased narrative ability.
The second step is to sit down and decide, once and for all, what magic can do, and what magic cannot do, and what it means to be magical in a world where everyone is magical. Quote:
Player's Handbook page 205
"The worlds with in the D&D multiverse are magical places. All existence is suffused with magical power, and potential energy lies untapped in every rock, stream, and living creature, and even in the air itself. Raw magic is the stuff of creation, the mute and mindless will of existence, permeating every bit of matter and present in every manifestation of energy throughout the multiverse."
In this context, saying that one character can do things "because magic" and another cannot seems a bit strange- it likely could be that anyone can perform feats that only magic can enable, under the right circumstances. Obviously, there are some who don't want explicit magical warriors flying through the air and launching energy blasts at one another, and that's fine- but some baseline of "at some point, we expect everyone, even the Fighter, to augment their natural abilities with innate magic to accomplish superhuman feats, once the game has reached a point where one would need to be superhuman".
With boundaries clearly set in place, then step three is simply to decide what sacrifices must be made to balance what a semi-magical character (as eventually everyone will be, even if not initially) can accomplish vs. what a purely magical character can accomplish.
Rather than lean on the traditional balances of AC and hit points, which I feel are woefully archaic, perhaps a system where one where spells have a hit point cost (call it strain, and have it be a kind of damage that only resting can cure, not magic)?