turntable motor speed

Yes, the change in speed in a DC motor is directly proportional to its voltage input: more volts = more speed; less volts = less speed. A 24VDC motor typically exhibits its nominal operating characteristics (speed, torque) when operating at its rated voltage. Too much voltage and the motor will be destroyed; too little and the motor will stall.

The regulation can take on many different approaches: since most brushed DC motors have mediocre speed stability, constant compensation (closed-loop feedback) is a must. This compensation can take the form of a tachometer which measures some point of reference as it moves by (either on the motor itself, or the thing (like a platter or pulley) the motor is driving) and adjusts speed according to how far off the from the reference mark the resulting measurement is - more points of reference equates to higher levels of accuracy. Another form of feedback measures the average current draw of the motor, and compensates for any deviation from this ideal. There are several other types of feedback mechanisms as well.

Brushless DC motors behave much like multi-phase AC motors, and require substantially more complex drive and feedback circuitry to accurately 'time' each phase.

Given the complexities of DC motor control, one can understand why synchronous AC motors remain a mainstay of turntable manufacturers despite some of their drawbacks.

No. This is nothing more than a lab-grade highly regulated DC power supply. It's the motor itself that varies in speed, regardless of the accuracy or "purity" of the incoming voltage; you can put a $10K DC supply in front of the "best", most expensive DC motor and it will still drift in its speed stability - not a lot, but enough to be noticeable. Such is the nature of a dynamic load, like a turntable.

While the current-sensing option on the Acopian supply supply will work, my guess is that it will maintain the output voltage within a range of +/- 2 to 5%. Not bad for picking widgets off an assembly line, but insufficient for controlling a turntable's speed. You need something with an accuracy of around +/- 0.1%.

There aren't too many stand-alone DC controllers for turntables. The Origin Live box works OK, but once you pair the controller and the box together you're already looking at $600 or so. Oh, and Mark Kelly made a nice DIY kit controller as well...

The above should have been 'once you pair the controller and the motor'.

No, the problem (for me) is long-term drift...such as the table running fast or slow up to 1 or 2% over the course of a side (we're talking simple voltage regulation-style control without a feedback loop). All those tiny errors add up. To me, the short-term, second to second errors of a DC motor are fairly benign, though certainly not insignificant when heard in the context of a good AC drive system, given its superiority of torque delivery, accuracy and so-called "PRaT" (and the spectral dividends that result from this approach). At least for me. YMMV and all that...

My experiences with various controller topologies for DC drive would imply a speed accuracy of around +/- 0.1% is achievable under ideal conditions.

Both systems (AC & DC) have inherent weaknesses and strengths; each listener has their own set of biases and expectations as well.