UC410/UC3410
claims it can support Logitech mice’s special buttons, do why does my Logitech G502 does not work?

You need to use Logitech setpoint software to do the setup for these special buttons, and for the Logitech G series mouse, you should use LGS (Logitech Gaming Software) instead of setpoint software, please refer to the below link: 

http://www.logitech.com/en-us/articles/11650