Linear Power Supply?

Rega tables have optional PSUs fairly down their product line and they are standard on their top 3 models I believe. The PSUs plug into a wall wart.  A rudimentary speed measurement (with an iPhone on the platter) showed RPMs of 33.28 and 44.98 (.15% and .044%). I suspect with that weight removed it would speed up a tiny bit, but I would challenge anyone to detect the difference.

I put a LPS into my phono stage and it probably made a difference statistically, but I couldn't hear it, but it wasn't an A-B comparison. That would be really tough to do. The manufacturer told me if it was his brother in law, he would tell him to do it. He didn't mention if he liked his brother in law :-)

So I purchased a SOTA LPS recently and I've been using it for a couple of weeks. The sound quality certainly isn't worse than before and it may well be better. It measures marginally better through Analog Magik. When I last measured Wow/Flutter in late 2021, I was getting .066 and now I'm getting .064. I doubt that I can hear a difference based on this alone. Also, I'm using the same belt as before, which may have stretched slightly - a good thing in this case, and the bearing temperature may have been different. I did let the turntable play about 10 minutes before testing. Also of interest is the possible reduction of switch mode power supply noise fed back into my system - even through a PS20, which I suspect may not completely eliminate such noise. On a side note, I have also eliminated all switch mode power supplies except for those on the ethernet side, which is isolated optically. So, maybe the SOTA LPS is better, maybe it doesn't make a difference. It's hard to know because of all the variability in the system. But based on what I've mentioned, I am glad I purchased it.

I am not knowledgeable enough to concoct an explanation for why a tube based motor controller might outperform a solid state one, for performing the function required in the Brinkmann system. 

You need look no further than the NASA Apollo space program.

They bought up most the the Marantz 9 tube power amplifiers when they were launched and used them for power controllers to provide fast power switching.

Similarly I had a friend who built massive mono blocks - 2kw transformers used for the input stages, 5kw transformers used for the output stages. He designed up a rectification circuit using 6550's because the off the shelf rectifiers were melting and could not switch fast enough.

I don't think the NASA story or the friend story are very relevant to either the question of why some llisteners claim to prefer the tube power supply for a Bergmann turntable over the SS alternative or to the question of using an SMPS vs an LPS to drive the Phoenix Engineering power source that in turn drives a turntable motor.  But I also don't claim to know whether or not substituting the SMPS for an LPS would make an audible difference.  Like I have tried to say at least twice, there is no predicting what listener bias can do to one's perception of SQ in that case. I do say that Bill Carlin is a very competent engineer and that I would trust him to design a very good SMPS for his equipment.