I have a few.
A PSA upper on my daughters 300 BLK that she deer hunts with. Its a 1-1.2 MOA for 10 at 100 yards with 125gr Sierra Pro Hunters.
Have a 223, that at this point is torn apart waiting on some parts. But it was a 1.5" gun with factory loads. But then figured out it was a 1-7 twist barrel that I was shooting light stuff through I think it didn't like (got the gun on a trade)
Also have a 6Arc that I built from scratch and still need to work up loads on it with the new barrel I just got. Hoping its going to be my deer/Coyote gun.
I Think any AR will do what you need. With the 5.56 you could probably search around any pawn or local gun shop and pick one up pretty reasonable then build from there.
That's what I did for my first one. They are a pretty reliable system and really easy to work on with just a few tools you probably already have