On the Symantec website, which you refer to, it does indeed state "Save your Product Key . . .".
However, on the next line, it then says "You may need your Product Key when you reinstall your Norton product".
http://service1.symantec.com/SUPPORT/tsgeninfo .nsf/docid/2005033108162039
You've no intention of reinstalling Norton, so you don't need the product key. Simply double-click the icon for the removal tool and follow the instructions. Then install your choice of security software.
Chris