The Germans also experimented with all-assault rifle platoons and attacks using full auto assault rifle fire of the attack element. Some enthusiastic officers even asserted that success in battle showed no need for machine guns on the attack any more.
These reports were from about the same time when more normal mechanized infantry squads were strengthened by a second MG42...

It appears that the effect of suppressive fires is first and foremost dependent on the enemy. Late 1944 Germans attacking Russian infantry in a local surprise attack probably met much more brittle opposition than was usual in other conflicts.

The post-war German army didn't even attempt anything like that, in fact it introduced the FAL and then the G3. These battle rifles were less well-suited for a high volume of fire and more generally combat out to about 200 or 300m (I forgot the exact figure from the relevant early 90's test which compared G3 and AKM) than the StG 44 and similar weapons.