the "eaton- coeey" conversion is absolutely safe and the original carcano was one of the strongest actions in an US army test after WW 2.
I think the idea is hilarious that also in the 1920s a well respectet firm would sell a hunting rifle that was unsafe in any way. the 6,5x54 Mannlicher Schönauer was not a modern pressure bomb and the craftsman who did this conversion understand very well what they were doing.
if you have a mauser 98 and a carcano side by side you see clearly the carcano action is a little bit narrower and lighter built for a smaller cartridge. perhaps not like a bolt-action rifle specifically for the .223 Remington, but somewhere in between.

nevertheless, it's no problem for the action to handle the 8x57 IS and the .30-06 if you drill out the barrel.iirc, the US army rechamber a 6,5 mm carcano to 30 06 springfield, fire this and the rifle survive. more is not possible.