The 5.56 v 223 was a real issue. I somehow doubt that 223 rifles made today have throats incompatible with 5.56 ammo. A huge problem was that the 5.56 tracer bullet is very long and a 223 throat is short. There was specualtion that the tracer bullet woudl sometimes expand in the base area before entering the case neck - causing serious problems.
I know one rifle manufacturer modified the 223 chamber to look more like a 5.56. Rifles are still marked 223.
As for 308 and 7.62 ammo, they are dimensionally almost the same and are functionally interchangeble. The differences are in the rifles, with 7.62 chambers being somewhat large compared to 308.
The pressure limits of 308 and 7.62 are about the same. the upper limit is around 60,000 psi, with most ammo averaging around 55,000 to 58,000 (of both flavours). There is some confusion about 308 v 7.62 pressure because the military specs were set with CUP units, but they referred to them as PSI, which makes their numbers look like SAAMI numbers. But, when you shoot them side by side in a pressure gun, you find the prssures about the same.
Here is some data. In this particular test, commercial and military match ammo was compared. The Norma match was rather mild, and the military a bit hot, with one round at about 62,000 psi. This is still acceptable and hardly unusual.