If you've got a friend who has an arcade stick of a different brand, have them bring it over and plug it in. That will rule out the USB ports if it works fine, and if it doesn't, I'd contact Sony.
If it works fine, then it's either the PCB itself or the USB cable. IIRC the Drone is a licensed product so it shouldn't suffer from authentication issues. If you bought it brand new from a retailer, I'd contact the retailer and have it replaced in that case. If you bought it used and the warranty's gone, I'd just crack it open and check for frays in the USB cable or a loose wire from the cable to the stick. It could be something as simple as the wiring harness that connects the USB to the PCB coming slightly loose.
And next step - simulate clicks via little sound system in stick. With adjustable volume control of course
As someone that switched to an optical stick about a year ago I can honestly say you won't miss the clicks.
Seconded. My concern was that I would want to hear the clicks too, but when I played with it at Evo, I didn't even notice or miss them. They were in no way my guide for finding my directions, and honestly I'm fine without switches now.