Getting PWM with an astable multivibrator (square wave generator)
I'm pretty sure that the only way to get PWM from this circuit is to have different charging and discharging rates for the capacitor C. You can change the operating frequency by R2 as you described, but you don't get duty cycle control.

Usually, you need two elements to make PWM - one to control the overall period, the other to control the on-time and off-time. You get this sort of configuration easily if you opt for a 555 timer-based PWM, which internally is two comparators with a latch and some other glue. You can get good duty cycle control with this approach, 555s are common, and the parts count is reasonable.