Unfortunately replacing the micro is not an option even if you knew the generic part number.
Before you junk the board just probe the pins of U1 and satisfy yourself that the device is getting what appears to be a sensible voltage on one or more of it’s pins, excluding pin 5, that could be VCC.
It could be 3v3 or 5V, I believe you thought the board ran from a 3v3 supply, check for this on U1.
If that looks ok then the processor port on pin 5 is probably defective and it’s either purchase a replacement board or junk the cooler.
To give you an idea of replacement cost ebay has a very similar board labelled Howden wine cooler display control board 082641452 for about $42.

I did see that board....but would it work with my machine? I guess it would right?

 

Unfortunately replacing the micro is not an option even if you knew the generic part number.
Before you junk the board just probe the pins of U1 and satisfy yourself that the device is getting what appears to be a sensible voltage on one or more of it’s pins, excluding pin 5, that could be VCC.
It could be 3v3 or 5V, I believe you thought the board ran from a 3v3 supply, check for this on U1.
If that looks ok then the processor port on pin 5 is probably defective and it’s either purchase a replacement board or junk the cooler.
To give you an idea of replacement cost ebay has a very similar board labelled Howden wine cooler display control board 082641452 for about $42.

What if I find on pin 1 another low voltage reading that is not 3.3v or 5v? Would that mean that possibly there is an issue with something else like a capacitor or something that I can replace and save this board?

 

The board on eBay is not the same as it has two JST type header connectors and component placement is different.
I don’t know where you are located but it would be worth at least trying to source the exact part.
If you can’t find a ’normal’ high level voltage on at least one of the micro pins it suggests there is something fundamentally wrong with a power supply.
There is a device on the top side of the board with 3 in-line pins that are soldered near R3, is this a three terminal regulator?

 

The board on eBay is not the same as it has two JST type header connectors and component placement is different.
I don’t know where you are located but it would be worth at least trying to source the exact part.
If you can’t find a ’normal’ high level voltage on at least one of the micro pins it suggests there is something fundamentally wrong with a power supply.
There is a device on the top side of the board with 3 in-line pins that are soldered near R3, is this a three terminal regulator?

I'm not with the board right now but I'm about 95% sure that's a transistor