That is the by product of instant gratification. Nature doesn't work on our time table, it has it's own. In the fall they will not have time to "sink in" and in the winter, just like the reason we own freezers, nothing decomposes, it is held in suspended animation. However, by late Spring or early summer none of those remnants are to be seen again. They have decayed, just like in a mulch pile and become part of the earths surface. Haven't you ever wondered how there got to be layers of soil over many years. How when you dig down you find definable layers of soil. That is how nature works. Give it the time and it will be self supporting. You can still add the chemicals that are part of Scott's to help feed the lawn, but, you will not need as much of it. Not chopping them up can cause a protective layer that will temporarily kill off plant life, but. in small pieces they will only do good.