It's not an old wives tale, it's about the chamber.
It's very unlikely you'll have a problem, UNLESS you have a short throat, and long/seated long projectiles, then you might get some pressure issues, if you do don't shoot that ammo.
I consider 5.56/223 interchangeable, BUT I don't own an older 1 in 12, 1 in 14 varmint gun, or a custom with a really tight chamber, if I did, I likely wouldn't be shooting 62gr/55gr 5.56x45 because it likely wouldn't shoot for ####, pressure signs, or not.
New rifle, and want to shoot 5.56 through it, fine, go buy 20 PMC 5.56 in the bullet weight you want to shoot, and try them, no pressure signs, you're golden, go buy a case.
As for the split necks, that's not due to you shooting "5.56" in a CZ, hard brass is usually the cause of that.