[COLOR="Blue"]Do you believe it could be something else?[/COLOR]
It could be just a bad wire but I cannot really say.

Unplug the unit.
Set your meter to it's most sensitive resistance mode/scale.
Short the meter leads together so you know if there is a zero offset in the meter.
Place one meter lead on the A prong of the power cord and leave it there.
Now measure to the brown side of the thermostat. It should be 0 ohms.
Measure to the yellow side of the thermostat. It should be 0 ohms.
Measure to the yellow (contact 1) on the defrost timer. It should be 0 ohms.
Measure to the gray (contact 4) on the defrost timer. It should be 0 ohms.

If all are OK problem is likely on the white side of then wiring.

[COLOR="Blue"] I opened it to see if the contacts were touching throughout the turn cycle and they are.[/COLOR]
Not sure if I am interpreting the above correctly but if one set of contacts stayed shut during a full rotation there is a problem.
One rotation is 24 hours.
So contacts 1 to 4 should be shut for 23.5 hours (cool time).
Contacts 1 to 2 shut for half an hour (defrost time).
The above are approximations as I do not have the actual timer specs.
Note that the 24 hours is compressor run time hours.

Below is a link to a manual.
It is too big to download to this site so I have had to use a third party site.
Un-click the Use Our Download Manager button and then use the grey button to download it, otherwise the site may put software onto your computer that you do not want.
Download KUWS24 Wine Cooler.pdf
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I apologize; I meant volts and not amps…my mistake.[/COLOR]
Not a problem.
Could you tell me how many volts and where you were measuring using the wiring diagram on page 7-1 in the manual for reference.

The condenser/compressor fan is not running at this time and the 'fan

took the defrost timer cam apart and it does turn.
Did you try just turning the cam?
There are two reasons why the defrost heater could cause a problem.
1. the the cam is not turning.
2. the contacts are not closing.

I am assuming that the interior light works.
If not check the wall receptacle for power.

Now there are two top of the list things that can cause the fans and compressor to not run.

1. The main control thermostat contacts are not closing.
Unplug the unit and one wire at the thermostat and check across it with your meter. It should read 0 ohms. If not it needs replacing.

2. The defrost timer contacts (1 to 4 are not closing when in cool mode).
Same type of check as as above but you have to rotate the cam through 360 degrees just to be sure you are not in a defrost mode when you check it.

FYI: the defrost timer motor gets power through the defrost heater and it's thermal fuse when in a cool mode. It also uses this circuit path when in defrost mode but only advances after the ice has melted off the evaporator coils and the defrost thermostat has opened.

Here are your parts. You were missing the last two numbers in the model number so it is for a 01 unit.
Parts for KitchenAid KUWS24RSBS01 Wine Cooler - AppliancePartsPros.com

[COLOR="Blue"]I checked the voltage for the starter and it is getting amps. [/COLOR]
I think the above may just be a terminology problem re: you mean volts not amps.
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I don't know if the starting fan and components have gone bad or the compressor and parts?[/COLOR]
The above is confusing as I do not know what you are calling a starting fan.
Perhaps the following will help.
Is the compressor running?
You can usually hear it but should also be able to feel it vibrating.
Is the condenser/compressor fan running?
If yes then the compressor should be running as they are wired in parallel.
So if it is on then the problem could be with the start device or the compressor itself.
Is the inside fan running?
It should also be on whenever the compressor and it's fan are on.

If neither fan and the compressor are not running then the problem could be with the defrost timer or the temperature control system.
Try turning the defrost timer cam (it only turns in one direction). If the unit fires up replace the defrost timer.