Unlike most on this thread I have actually done this. I used a VPI classic 3 and had some slight motor noise when up close, with some associated vibration, I got great sound and stable speed. When upgrading power cables I had an extra RSA Mongoose and plugged it into the table to see if it affected the motor noise. It was clearly audible and reproducible. I had a lower noise floor when I turned up the volume with the table spinning and no stylus in the groove. There was no audible improvement when I added the SDS, adding another Mongoose to the SDS eliminated any remaining vibrational motor noise. I clearly and conclusively hear this effect on the noise floor and ear to motor, whether I could double blind the table with music playing I can't say with any confidence. Knowing the motor noise and subsequent vibration are likely altering in small ways the stylus motion, I kept the cables in place and was happier with the sound. Especially on LP's like astral weeks that I enjoy loud, they have lots of quiet parts where the lower noise floor shows itself. When I upgraded to the Eagle and Roadrunner, I thought it might be silly to reuse the RSA Mongoose cables as there were skinny cheap looking power cables in the same current path(The eagle has a captive low gauge plug). The motor noise returned with stock cables, I went back to Mongoose's. My sound quality improved and motor noise dissipated again. The eagle/RR also improved my TT sound, an effect the SDS didn't have, independent of cables. So I do believe a TT power supply may affect sound as well. Though in that case it might be more of a speed related issue and not noise. To be clear I have no evidence that the cables on my power supply affected the speed, though the displayed speed on the RR definitely seemed more steady , that is purely anecdotal.
Why is this occurring and make no mistake it is, at least in my system

? I think Tim A hit a key point about a better cable allowing much less motor noise to be transmitted into the line. It blocks noise coming from the wall and back to the wall better than the standard cable. IMO a lower noise floor. Remember with AC it blocks noise 60 times per second(roughly). I think it's that simple, less noise and vibration. I was alway very skeptical of power components and now utilize some costly models. It is easy to try yourself, plug a new cable into your TT, put your ear to the motor on 33/45 speeds(as I had much more noise at 45), then switch cables I bet you'll hear a difference, not necessarily better, but if you have noise at the motor I think you will be shocked what a cable can do. The next thing is to crank your system with phono selected and the arm over the table with it spinning, do this with both cables. For me I have a hum that starts around 12 o'clock(crazy loud for Naim), with the stock cables on the TT and power supply it started to intrude around 9 o'clock. With my VPI the high end cables were a winner.
I now upgraded to a Clearaudio Compact Wood which ironically has a cheap wall wart and dc plug. It smokes my VPI. There is a second party power supply upgrade for the clearaudio, they want 1k for it so I'm in no hurrry, but I have no doubt it may improve my slight motor noise and improve the sound.
Jeff