Since it looks like we're not worried about compatibility anymore . . .
In other words, a system based on 5E that only removes or simplies from 5E, doesn't add anything new.
If you change how systems in 5E work or add new things, this is no longer stripping 5E to its core. So, for this concept, while many of the suggestions upthread are great ones, they are changing or adding, not stripping down to core.
I'll tear some stuff out of the 5RD. See if we can get down to the core, and still feel like D&D. You might want to duck, I'm just gonna throw these over my shoulder:
Advantage, Dex, Con, Wis, races, classes (and subclasses), alignment (see below), hit points, AC, attack rolls, saving throws, damage bonuses, proficiency, time, movement and grids, environment (includes light rules), crits, action types, death rules.
ABILITY SCORES & MODIFIERS
RACES
CLASSES & SUBCLASSES
HIT POINTS
SKILLS & PROFICIENCIES
EQUIPMENT
ADVENTURING
COMBAT & ACTIONS
FEATS
Okay, I think I covered most of these. And threw most out.
Why keep Inspiration? Because if you look at it as "you did something cool" instead of "you get a bonus to a roll," it's actually a really fun, simple, rule.
Initiative stays, but gets a twist. No one wants to get the last turn. The whole battle could be over by then! Roll initiative, and your result determines
your priority, not your turn. So you can act whenever you want, but if there's a conflict with another character, initiative determines who beats the other to the punch. This rewards indecisive PCs with later-round-actions.
Let's keep ability
scores for Str, Int, and Cha. They're not useless - especially with Hit Points removed from the game. They'll determine how much damage you can take. Which means we'd better keep ASIs, because there's gotta be something to award at level-up!
Alignment's out, but Flaws, Traits, Bonds, and Ideals remain. Alignment complicates D&D (despite its severe gutting), while the FTBI do something much more useful: make characters interesting without complicating rules.
Does it still resemble D&D? Let's look at the demos:
Fireball
3rd level evocation . . .
Hmm. Fireball doesn't need to be evocation. Maybe not even 3rd level? Let's get rid of most of the magic rules.
Fireball
1 action
Range: 150
You cause d8 heat damage to one target.
Short and sweet. Without saving throws for monsters, you'll have to roll a d20 test (the only type of roll left) to see if your fireball hits. Without Wizard Levels, how does a PC know she can cast it? We'll have to add a difficulty to spells, so the caster will need more ability to cast higher-level spells.
Fireball
Level 1 (1 action)
Range: 150
Difficulty: 0
You cause d8 heat damage to one target.
There we go. It's obviously first level since it only does d8 damage, so we'll keep the difficulty easy. The "wizard" will get a bonus from his Intelligence score to offset the spell's difficulty. We're keeping skills too, so that will provide a bonus as well. It's not hitting everything in a "20-foot-radius sphere," but a level 2 fireball could easily hit "two targets." Or cause more damage to one target.
Let's look at the orc berserker.
Orc Berserker
CR 1, 200 XP, Prof: +2 . . .
That's not feeling "core" to me. Let's replace all of that.
Much better. It will need some abilities, skills, and gear. A description would be nice, too, for those who don't know whom orc berserkers are.
Orc Berserker, Level 1
A big, burly, axe-wielding maniac from the lands of Orsinum. Ideal: chaos reigns! Trait: berserk.
STR 14 (+2), INT 8 (-1), CHA 10 (0)
Skills: attack 1
Gear: ring mail d6, greataxe d10, javelin d6
With hit points removed, the orc's physical damage capacity equals her physical score: 14. But, no AC? The ring mail reduces damage instead of reducing odds of a hit. Instead of a STR bonus to damage, the orc can upgrade from a d6 handaxe to the d10 greataxe. STR still adds to the test for attacks. She doesn't need a speed rating either, since grids are out. If there's a timing question ("did my halfling outrun her?") the DM can ask for a STR contest.
This is important information. Seriously, I'm not kidding nor making fun of it. The d12 is among the least used polyhedral dice in D&D mechanics. But still loved -- so we can't drop dice types and still have it feel like D&D!
Agreed - the d12 is loved, and is least used. Since I tossed out damage bonuses, die-type increases fill the gap of Hurting More. As with the greataxe, heavier or more dangerous weapons use higher die-types, and the d12 is often the top of that food-chain.
By now, I've probably broken 5e pretty well, and left a couple of mechanism gaps that need filling (TotM positioning, feats). I hope, however, that I've provided some ideas on what's the "core" of 5e, and what this thread's work-in-progress might consider. Here's a possible conclusion of the above modifications, with the gaps filled in to make it fully playable:
www.drivethrurpg.com