On the other hand, we also need to ensure that burst or auto fire do not "break" the game by becoming too powerful.
That's why we have weapon balance. It isn't about the rules of single fire/burst fire/auto fire, but the weapons that use them.
This is entirely backwards, in my opinion. The weapons should be designed to function within the combat system, not the combat system built around the weapons that exist. Your cake thing makes no sense, and I'm not sure you've thought through what having autoguns inflict 2d10+8 would do to the rest of the system.
My point is that each of the rates of fire should have a reason to be used. Full auto might be the best choice at point blank range, but at 3x normal range? You're going to want a precise shot or you're not going to hit anything. The rules should provide a framework that give players meaningful choices to make in combat, not just, "Well I'll full auto at him again because that's always the way to go." That'd be like fighters full attacking in 3e, which was boring and unfun.