Safety.
If your prop governor fails in flight, you can still fly the airplane.
If a full-feathering prop fails in flight, you cannot fly with it feathered.
If your engine fails, you are going down anyway. Fine pitch reduces your glide, but not so much that under ordinary circumstances you cannot pick a place to land. It could even be a benefit, helping you slow.
I've never heard that full-feathering, or even Beta-capable, props are heavier. They might be, but I don't believe it has anything to do with why single engine planes are generally equipped with props that fail to fine pitch.
Twin engine airplanes often have props that fail to feather for the same reason - safety. If one engine or one prop fails, you can still fly the airplane.