Case in point: the M14E2 or M14A1 is the selective fire M14 dressed for the automatic rifle role (suppressive fire). To do this, the M14 had its selector lock removed and replaced by a selector switch, a new stock with a pistol grip, hinged butt plate fitted, a folding front grip added, and a muzzle compensator slid over the flash suppressor. The setup was similar to the old M1918A2 BAR of WW2 and Korea, but slightly lighter. In trials against the belt fed M60 GPMG, the M14E2 or M14A1 could not compete for sustained fire or accuracy. And so these automatic rifles were reconfigured to standard M14’s with their selectors locked.
If you are going to suppress the enemy, then you need a good belt fed machine gun with quick change barrel capability. The M60/M240 or Mk 48 Mod 0/1 work well in this role. If you want any kind of control with a 7.62 NATO caliber rifle in full auto, that means a beast approaching 15 pounds or more. But, the draw backs are that its magazines of 20 or 30 rounds limit it (frequent changes) and it overheats. In 7.62 NATO caliber machine guns or rifles, there is no free lunch.
To be full auto controllable in 7.62 NATO = gun of about 18.5-27.5 pounds, belt fed, and with a quick change barrel. Automatic rifles that are merely infantry rifles in 7.62 NATO with a selector switch are not controllable or accurate in full auto.
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.