I think there are other ways to obtain control with automatic weapons than merely adding weight to the gun.
The BAR was 19 or so pounds, and had the reputation of being too accurate with automatic fire from the bipod. Its slow rate of fire, and the bipod location on the end of the barrel tended to reduce the error induced by fire.
The WWI 8mm Lebel chauchat had a very low rate of fire, and its long recoil system spread recoil force over the entire operating cycle. A properly designed long recoil system with a low rate of fire can have the ability to suppress with good accuracy, so long as there is no time for an enemy to pop up and shoot between bullets. The .30/06 Chauchat didn’t work. There was a reason why the 8mm Chauchat was the most produced MG of the war, and served afterwards in the ‘Pygmy Wars’ that occurred after WWI was over. Because of its low rate of fire, it didn’t overheat very quickly, though it would overheat.
Suppression occurs in part in the mind of the opponent. You can’t suppress a lion, nor a committed jihadi. Better for each to give them multiple solid hits that diminish their ability to perform combat.
The standard select fire FN FAL or M14 weighs in at 12 lbs and can't hit the broadside of a barn on full auto. The US Army spent millions of dollars in the late 1940s and early 1950s trying to prove it could and the laws of physic proved them wrong.