I have just acquired a 2nd hand fridge that doesn’t appear to be cooling at all. It is making a huming noise when switched on, and the condenser is getting warm so obviously it’s trying to do something, but the coils in the top of the fridge area are not cooling at all.
The fridge runs fine, no clicking, no shorting, makes a soft inaudible humming noise (can only hear when pressing an ear against it) and all the lighting etc is working fine.
A pinhole leak over the years could mean loss of coolant.
BTW, does it cool at first then after a few weeks it stopped? Are you sure it’s not that the timer is gone, the fridge is stuck on permanent cool, and never goes through the defrost cycle and the cooler coils are all iced up? I had a fridge like this. Let it sit off and open for a few days, all the ice melted, and it worked OK for a month or so then gradually stopped cooling. (Or open it up and take a hair dryer to the cooling coils under the freezer compartment)