This does make me wonder what a FATE: Accelerated version of 5e would look like...
If I am remembering right, FATE Accelerated just had abilities and some powers you made up, no skills.
Close, but not quite. But yes, I think you could FAE-ify D&D. (FAE is the standard abbreviation for "Fate Accelerated" - not that E is prominent in the name, but it makes it easy to say).
FAE has "Approaches" instead of stats - Careful, Clever, Flashy, Forceful, Quick, and Sneaky. You could swap those out for the D&D stats, or not, I suppose.
FAE then has Aspects and Stunts, which together cover all the areas of class abilities and skills. Each character specifically has a "High Concept" aspect, which tells you in a few words what the character is. For D&D, you can pack a lot into the High Concept - race, class, subclass, and even some very basic background can fit in there - Half-orc Totem Warrior of the Hokey-Pokey Clan, for example, is a perfectly good High Concept Aspect.
Aspects are great for broad classes of things you want a character to be good at occasionally (you have a spend a Fate Point to use them). Ranger of the High Hills as an aspect means that you can pull out a wide variety of rangery stuff now and then.
Specific skills and special abilities that you want to use frequently fit into stunts FAE has two forms of stunts:
1) Since I <have quality> I get a +2 whenever I <Action> with <Approach> in <circumstance>.
2) Since I <have quality> I can do <special thing> in <circumstance> once per session.
So, for example:
Since I am a Master of Melee, I get a + 2 whenever I Forcefully Attack in melee combat.
Since I am a Military-trained Sniper, I get a +2 whenever I Carefully Attack with a firearm.
Because I am a Trained Thief, I get a +2 when I Carefully Overcome mechanical obstacles (like locks and traps).
Because I am Quick on the Draw, once per session I can choose to go first in a physical conflict.
FAE isn't great for managing long lists of detailed spells, but classes of spells become simple:
Because I am a Fire-Evoking Wizard, I get a +2 whenever I Cleverly Attack with Fire Magic.
This would wrap all your fire combat spells in one big bundle.
Because I am a Bard of the College of Glamour, I get a +2 when I Flashily Overcome social obstacles.
Because I am a Cleric of Life, I get +2 when I Carefully Overcome to remove stress and Consequences.
Or class abilities
Because I am a Druid of the Moon Circle, once per session I can turn into a bear.
And so on.