Linear Power Supply?

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.

The LPS perhaps keeps the platter spinning a bit longer because its capacitors hold charge for several seconds after unplugging the unit. Until the DC voltage falls below a certain threshold, the platter will keep spinning. No magic.

I'm sorry.  Where did I "deny" anything?  I simply pointed out that the OP is concerned about replacing a wall wart that contains an entire power supply, which launched a discussion of switching power supplies (SMPS) vs "linear" power supplies (LPS) whereas your wall wart was (apparently based on what you wrote) a transformer.  Whatever else constitutes the power supply of your turntable must reside on the TT chassis; it could be an SMPS type or an LPS type.  Changing the transformer does not change the downstream characteristics of your PS, except whereas many times a "better" transformer may favorably affect function, at least to the ear.

Viggen, Sounds like in your case the wall wart WAS a transformer, only. That’s different from the situation under discussion where the wall wart is a complete power supply, meaning a transformer plus the downstream parts required to generate DC at a particular voltage and current.

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.  But I do know that such a rationale exists somewhere, if not on the Brinkmann website.  Several years ago, Mark Kelly, a very smart super-hobbyist, developed a tube based motor controller for the Garrard 301 motor.  Mark could give you reasons why he chose tubes for the application rather than solid state devices.  Prior to developing his tube unit, he had already perfected (and sold several) an SS motor controller for the 301.  He built maybe a dozen tube units before going in to business building and selling wood bicycle frames, in Australia.  But it ought to be repeated that none of this has any bearing on the Eclipse system, where the issues are different, and we are only talking about a PS, not a motor controller.

Like I said, it's a matter of individual perception.  Is there likely to be a measurable  advantage to the LPS vs the existing SMPS, in terms of noise in the audio signal or speed control accuracy?  No, I don't think so. But will you hear a difference? Maybe, because of how our brains work.  Brinkmann has for years offered a tube-based motor controller as an optional extra cost item compared to their standard solid state motor controller.  I can't think of a reason for it, but many end users report an improvement with tubes.  (The present situation is not analogous, of course.)

You have asked a very complex question, believe it or not. First, one can argue whether "noise" from a turntable motor power supply can even enter the signal pathway. The answer is it can enter by direct radiation of EMI and/or RFI from the PS itself or by retrograde conduction from the SMPS into the power cord into the AC supply which could then contaminate the PSs for your amplification gear, if all the gear shares the same AC source. But in your case, you are protected from the latter phenomenon by the fact that the outlets of your PS Audio power regenerator are isolated from one another. (I trust you do not plug anything else into the outlet that feeds the wall wart.) Distance is the best way to avoid contamination by radiated EMI and RFI, in this case, since it is not practical to add shielding to a wall wart. OK, then the next issue, I think, is that not all SMPS Wall Warts are created equal. Surely some are better engineered than others, and I would feel confident that SOTA used the brains of Bill Carlin, the Phoenix Eng designer, to qualify the existing wall wart. So, let’s assume you have a "good" SMPS wall wart. (The PS Audio regenerator makes it even better because it cleans up the AC going in, in case the filtering of the WW is marginal.) To answer your specific questions, the PS serves the Eclipse/Phoenix Engineering electronics; it does not directly serve the motor per se, and I would guess that if the current setup is giving you stable speed at "33.33" on your Roadrunner, I doubt you would see any change in function with a better LPS; nor would it seem to run "more precisely", whatever you meant by that. I also doubt there would be any change in "motor vibration" on any measurable level. However, the brain is a strange place, so you might "hear" a difference. I have been using an Eagle+RR to drive my Lenco motor, for a few years now. I am completely happy with it, and I am not even sure whether it is running off an SMPS or an LPS. If I wanted an LPS, I would find out what voltage and current output are needed and then build it myself.