If you plan to use the just finished hot stock to go right into a soup pot, you do not have to cool it first.
Cooling it quickly is important as you know. Cooling it also solidifes the fat on the surface so it's easier to remove.
Any hot food should be cooled quickly to avoid longer periods of time in the bacterial danger zone of 40F to 140F. So, after you make the soup, it should be cooled quickly and refrigerated if you are not going to serve it right away.
"If you want to make an apple pie from scratch, you must first create the universe." -Carl Sagan