Linear Power Supply?

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.

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 

If you cant understand the difference between how tube and solid state rectification behaves then you probably should not be modding your equipment. Their conversion and the way they release of energy into the circuit is completely different.