millercarbon: Never misunderestimate the ability of the forum to overthink even the simplest things. Full disclosure: I converted AC to DC battery, did none of that, and it was audibly quite a bit better. Just not that hard. Unless you make it so.
Truer words were never written. If the Clearaudio table uses a DC motor, it is not driven directly from the wall wart power supply. At 33 RPM, it is most likely driven by 3-4VDC and from an internal controller which not only regulates the voltage but has active current feedback (negative output impedance) to compensate for the motor’s negative-slope torque/speed curve, without which, the motor will slow under load.
Regulating an AC-to-DC supply is done to remove any AC ripple from the output. The turntable should work just fine from an unregulated battery supply as there will be no AC ripple present. Regulating a battery supply in this case would do nothing other than complicate the application and waste heat and power. The rest of this discussion is just noise.
FYI, MC7812 regulators have a dropout voltage between 2.1 and 2.4V, depending on the device, so you would need 14.4VDC or greater for proper regulation. LDO regulators (Low Drop Out) exist with dropout voltages as low as 200-300mV, but they can become unstable and oscillate without careful layout and compensation.
Advice to the OP: As millercarbon alluded to, don’t over think it.