I was looking at the sound module website and if I understand it, the audio output power is "only" 1/2 Watt? I realize with the proper speaker and such 1/2 Watt can sort of fill a room...but for simulating a turntable I'd think you'd want to feel the groaning and moaning of the mechanism?
This goes to issue of volume. There are audio amplifier ICs/chips/modules where volume is proportional to a voltage - a feature called "DC volume control" for obvious reasons. I didn't find any low-cost (under $5) amplifier modules on eBay with this capability...so you might comment on how fat your wallet is.
I'm assuming you don't want some contraption where the volume control knob of some audio amplifier is somehow coupled to the speed-control shaft/knob of your turntable motor controller.