Yes, PWMing /G is the standard way to apply common duty cycle control to a TPIC6
x595. Don't forget its active low so you need to invert the PWM either is software or hardware.
The front page of the datasheet says:
Separate power ground (PGND) and logic ground (LGND) terminals are provided to facilitate maximum system flexibility. All PGND terminals are internally connected, and each PGND terminal must be externally connected to the power system ground in order to minimize parasitic impedance. A single-point connection between LGND and PGND must be made externally in a manner that reduces crosstalk between the logic and load circuits
which I would interpret as its forbidden to switch PGND (as it must remain connected to LGND). There's probably only a diode junction isolating the two grounds which is fine as long as there's no more than a couple of hundred millivolts peak difference between them. You are lucky the chip survived as uncontrolled current flow in the substrate of a CMOS chip can trigger latch-up.