I think you have a brilliant idea there - maybe you should think of patenting it, with something like Minced Mice Morsels.
My cat certainly eats the mice she catches. I think maybe it depends on cats' early experience - Our cat adopted us when she was about 4-5 months old, and we think she was either a ferral cat, or a kitten that was abandoned because her owner didn't want her. So mice would have been her standard fare. She is certainly not underfed, rather the opposite, so it is not hunger which drives her to eat them.