Luftwaffe insisted on 7.92X57 because of the experience of the Fallschirmtrupp on Crete in 1941.
When they dropped on Crete, they largely got seperated from the cannisters that held their carbines and MG's, so they were limited to pistols and a few subguns.
The defenders were out of range of 9mm, and the Krauts got chewed up pretty bad because of it.
So when it came time to finalize the cartridge for the Fallschirmjagergewehr, the paratroops themselves insisted on the standard full power cartridge in remembrance of Crete.
They wanted the range.