Are there any benefits in refrigerating stock before using it?
Any gelatin in your stock will be there whether it is warm or cold.  It's just that gelatin solidifies when it cools...it will melt out when you warm it again, leaving you where you are now.  You can go straight to your sauce, your stock is probably as flavorful as it will be when you are finished cooking it.  Cooling prior to refrigeration is to prevent the potential warming of other items in your refrigerator, and has little to do with the stock itself.