Whether it's a BS bigger flip, a bigspin, or a 180, pivot always helps.

You could either force your board to turn by sending your back foot forward or let the physics do the job.

Is it really because of your shoulders' rotation that your backside 180 is not spinning?

A backside 180 is actually composed of two elements: "the lower body's rotation that follows the upper body's preceding rotation" and "the lower body's rotation in reaction to the upper body rotating in the other direction." Combining them will help you land your backside 180s more easily.