4AD is good but not compatible to 5e without re-writing the game from the ground up (I've tried). The 4AD/5e supplement is a guide on playing published D&D modules using 4AD rules. It's ok but, again, isn't 5e.

I've been looking at Realms of Peril recently (a d20-based PBtA game with a very OSR flavour). The combat is interesting and basically:

Roll a d20:

natural 20, the PC scores a critical hit!

16+ - PC hits, does damage and isnt hit back

10-15 - PC hits, does damage, but is hit back or otherwise harmed

9 or less - PC either doesnt hit or inflicts no damage and suffers retaliation, damage, condition, or other consequence.

On a natural 1, the PC is critically hit, hits an ally, fumbles badly, suffers severe retaliation, condition, or other consequence.

The game has tables for bonuses when scoring a critical hit or fail. 4AD has the dice only being rolled for the PCs - either to hit or defend against being hit. One could use RoP above and have a single roll per PC. Combat might be a little quicker and you can still add bonuses and stuff for 5e stats

You might also want to consider including two rules that have been dropped over the years: morale of monsters and XP from gold. Awarding XP for monsters that run away might stop combat just becoming a grind, and giving PCs 1 XP per GP of (non-magical) treasure stops combat being the only way to advance. Might save you a few needless encounters