IIRC, the "can't shoot 5.56 x 45 mm in a .223 chamber" was started when folks tried shooting the military ammo in a .223 Thompson Center Contender. No one was specific about what happened, but TC took out print ads advising against the practice.
The linked article has what I expect is a slight error in the genesis of the .223/5.56 x 45 mm round. Some years back there was a long article on this subject in the American Rifleman. Remington (and Winchester) had been working for some time in the 1950's on a .22 center fire cartridge for the DOD Light Rifle Project referred to in the article as CONARC. Remington won the cartridge design competition. Which rifle to adopt remained open.
The Pentagon then shelved the Light Rifle project, probably Congress driven due to recent/pending adoption of the M14 & the fight for NATO standardization of the 7.62 x 51 mm cartridge. I'll skip the rest of the political high jinks that got the AR/5.56 into military service.
At some point during that skulduggery, Remington apparently-per the linked article- decided to release the .223 as a commercial cartridge, possibly/probably anticipating the sales surge that results from military adoption. (Also cutting the legs out from under their .222 Magnum and later, the .222.) But, the .223 really was NOT the parent cartridge of the 5.56 x 45 mm.