Great question, I probably will not do
enough justice to
answer your question but hopefully I can show a few angles to consider.
Chop and drop has its pros and cons. One con is that it can be unsightly while it decomposes and gets intermingled with the soil. I might mow my
yard and side chute a fair amount but sometimes there is TOO much grass and matting starts choking out growth. Could I shake it around and try to do a big even application? Maybe... But I am a lazy gardener!
Composting allows a product to be made of a variety of materials that contain varieties of NPK, minerals, microorganisms, and other goodies. Balancing your carbons to your nitrogens will more than likely require more than a small lot generates. I find my compost piles to be a great waste stream diversion from my household which now encourages growth when applied to plants.
One of my first Permie-ish
books that I read was
Compost Everything and it touches on not only the fertilization benefits but the tilth benefits compost gives soil. Compost as organic matter has a positive impact on garden soils. If compost is hot processed, you can introduce the compost without the worry of weed seeds which chop and drop might not if you are too late to the chopping.
There isn't really a 'right' way to do anything, but the variety of things you can do is pretty vast!