"Weighing and comparing 2 cases does not prove anything regarding wall thickness."

It absolutely does. Two cases of identical weight also have exactly the same internal volume. Lighter cases have more internal volume than heavier cases. Think on it awhile - you'll figure it out.

Part of the pressure issue of .223 vs 5.56mm involves the throat dimensions. The leade of the .223 chamber is longer than the standard military 5.56mm chamber, and the bullet gets more of a running start and the peak pressure is lower. That is also why the peak chamber pressure in a military barrel is higher.

By the way, the methods used to determine the peak chamber pressures of military 5.56mm and commercial (SAAMI) loads are different. The NATO EPVAT pressure measurement method used for the 5.56mm measures pressure at the case mouth.