Reducing speed of motor in cheap record player
[edit 1]
Note: as JRE points out in a comment, if the unit is still under warranty, try that route first. If so, see if you can reverse your potentiometer mod and bring it back. Even if there is no warranty, you may want to get an estimate on a proper repair. Then you can decide if you want to customize it.
My rule with taking apart electronics is that I will attempt to repair anything (excluding when there's a risk of fire or electrocution) as long as the owner already considers it a write-off.
[edit 2]
I hadn't initially looked up the unit. As JRE and Bruce Abbot have said, it should already have a PWM controller of some sort. It would need to be a ridiculously cheap unit not to. It'd be possible to use a potentiometer as a simple voltage divider but, with the low cost of PWM controllers, that seems unlikely. Even in a rock bottom turntable (which this is not).
You may still need to add a PWM controller, but only if the original one is broken. And that would also require an unmodulated 12V source.
So, assuming return isn't possible and repair isn't reasonable, the value of my answer right now might be more in determining what you have. I couldn't find anything online about the guts of that thing. Do you have access to an oscilloscope?
Even without a scope, there are still some things you can test experimentally.
Is that pot doing anything?

Since it's already connected, start with your 100k pot.
Crank it to
one extreme
Put a marker on the platter and Count revolutions over a
reasonable interval like 3 or 5 minutes
Crank the pot to the other
extreme and repeat.
Depending on your results, you may feel confident
to shorten your counting interval.
Reconnect the original pot and try
again.

When stuff like this fails, it's usually something mechanical. E.g. Ribbon cable pulled out of a socket, cold solder joint, missed solder joint, broken wire.
[Original answer]
PWM for Speed Control
If the motor has only 2 wires, I'd recommend a pulse width modulation (PWM) motor speed controller. Google &quot;12V PWM motor controller&quot; and you should get lots of hits. PWM works by briefly turning off the power for some percent of each cycle. The &quot;pulse width&quot; is the amount that it is on for each cycle. It does this really fast (should be &gt;20kHz) so the individual pulses are not detectable. Fan controllers are configured for 25kHz so they are outside of the range of human hearing.
You'll should find reams of information if you replace &quot;motor&quot; with &quot;fan&quot; in your Google search. People with pumped up computers tend to have a lot of 12V fans and an obsession with controlling them optimally.
If you buy from somewhere like Amazon, they're probably at least a few bucks each. They'd be cheaper from somewhere like AliExpress.
Alternative Wiring Possibilities (3-wire and 4-wire)
If the motor has more than 2 wires, then it likely has a tachometer of some sort. You can still drive it with the same cheap PWM controller but you could get better accuracy with a device to adjust the PWM based on measured speed. You'd likely need an oscilloscope or a meter that reads PWM to figure out exactly what it's doing.
3-Wire
A simplified model of 3-wire is that the third wire emits pulses per rotation (probably 2) then a controller reads that information and adjusts the drive voltage to adjust speed. Reality is a bit more complicated because the power source for the output is the power for the motor/fan. Since the power source is modulated there will be times when a pulse will be missing.
4-Wire
4-Wire is like 3-wire but, instead of controlling the speed by modulating the voltage used to drive the fan, it has an extra wire that gets fed with a control signal. The control signal adjusts the speed and the drive voltage can stay at 12V.
Have fun and let me know what you figure out. :)