The whole .223 / 5.56 incompatibility thing is only relevant in two ways as far as I am concerned:
1st: If the weapon in question is rather old, like Vietnam-era hardware in either caliber which back then, some of those guns had a variety of issues with their proper functioning which might make a difference in the incompatibility area.
2nd: If you have a fully-automatic weapon and are using 5.56 in a .223 weapon or visa-versa, then the high rate with those differing chamber pressures can cause more FTE’s maybe, or cause more chamber wear.
Not really a big deal with all of the modern weapons today from makers like Bushmaster, Rock River arms, Stag arms, etc… since all of the people buying these are buying semi-autos and will probably never fire enough rounds through the gun to appreciate any issues with the incompatibility. I own an after market M4 in .223, and I’ve fired maybe 700rds of 5.56 R.O.R.G. M855 through it in 4 years, and never noticed a problem. If I were a match shooter with a bolt action in either caliber though, I would definitely want to stick with the exact right caliber since those types of guns are very fine tuned, and might notice a difference.