Why do major Appliances go bad at the wrong time?

Our 11 year old kitchen refrigerator decided to quit cooling. I have heard clicking sounds from it for the past week or so. It was cooling okay. the freezer temp would go to about 12 and then it would cool back to 0. Almost every day.
Saturday I decided to vacuum the cooling coils. They were caked with dust. I don't know if we ever had cleaned them. After I did that and also pulled the back cover where the compressor is, cleaned there. Plugged it back in and it would not cool, freezer temp went to 29 and stayed there.
Fortunately we have a 2nd refrigerator in the garage that only has beer, wine and soda in it. Moved all of the food out of the kitchen refrigerator and freezer to the garage.
After searching on the internet I found articles on the clicking noise. They mentioned the Start relay on the compressor. I found where it was, pulled it off of the compressor and ordered a new one.

The three pins on the compressor is where the relay plugs in


The start relay is the grey item on the right. The black box is the start capacitor.


Hopefully on Tuesday I should receive the new start relay
If it works we just saved the cost of a new refrigerator