Then these paths you will be mulching are IN the garden itself rather than outside the bed. That makes a big difference. I thought you were talking about mulching paths between or around beds.
In that case I wouldn't put any of it in the garden. Just stockpile it and let it age for a year. Stir some alfalfa pellets in with it to speed it up. You can even cover it with black plastic if you want but you risk it turning anaerobic and killing off beneficial microbes as HE said above. So I wouldn't. If I had to for some reason 'd use perforated landscape fabric instead. Better to have access to air and water, and stirred/turned as much as possible.
As for fixing your soil. Don't put all your eggs in this one basket because this will be a slow process and there are faster ways to do it. I won't go so far as to say that tilling is what killed your soil. Many folks till without killing their soil so it isn't quite as evil a process as some maintain.
And it takes a combination of many things to kill soil as you listed in that other discussion. But the fastest easiest way to fix it is to mix in 6-8" of fresh multi-component compost (preferably with some manure mixed in it). And unless you have an industrial sized compost organization going you can't make enough in a reasonable amount of time. So find some place to buy some if possible. You can start applying it right now in between the rows and around all the plants without any worries of nitrogen binding on anything else.
Dave