Its not that they were too heavy for the SAW role, although I wouldn't like to be lugging them around at 11-12 kg + ammo, its that there really wasn't a SAW role in German infantry doctrine of WW2.
As I understand it, German infantry tactics were based around the machine gun acting as the foundation of the infantry killing power. The basic formation was a 9-12 man unit, comprised of a three to six man MG squad and a five to six man rifle squad. The infantry unit was basically there to support the MG's base of fire, keep it protected and fed with ammunition.
German paratroops, emphasising infantry movement more, did get a kind-of SAW, in the form of the FG42.