I own 2 side by side refrigerators. Both have a coil of water line in the fridge section. See below pic.
My guess is this coil is very long to allow lots of cold water to be stored in the fridge and dispensed at a single serving.
If I wanted to serve beer from the water dispenser I would cut the line already in the fridge and connect to a liquid quick disconnect which then connects to the keg out. I would keep the total length of water line to 10-12' max (much less than the length of the coil in the back of the fridge).
I am not sure of the inner diameter of the water line, but, if it is not what you need I am sure you could buy an adapter. This may be another issue if the line has large diameter. In this case you could use a longer length or place a stir stick in the dip tube to try and decrease foam.
As some of the line may run outside of the cold provided from the fridge you may end up with foaming problems.
I like my taps better. I say save the water dispenser for water for when you need it in the brewery.