Yes it will temporarily tie up some nitrogen as it breaks down but it is worth imo. I'd go with rye and red clover off the bat. You could even spring plant it. I've been suggested many times.
moisture and nutrient retention are 2 simple ones. You gotta understand the synthetic fertilizer you buy isn't organic. A plants roots can't just grow around a piece of pot ash and capture the potassium. That pot ash needs to break down and chemically become organic, so a plant can use it. It is a very inefficient process, so your plants are only getting a fraction of the expensive synthetic fertilizer you are buying.
Organic matter is criticial in holding and making organic nutrients available to a plant, reducing the need for the expensive organic fertilizer.
It all comes back to $ my friend, and how much you are literally throwing away.