Why doesn't Nehushtan require its wielder to sing?
The answer turns out to be simple. There are two key facts here:


Nehushtan is a complete relic. I'm not sure if this is mentioned in the show, but it is mentioned in the season 1 keywords. 
According to Fujitaka in episode 3, a complete relic need only be activated by song a single time, with the caveat that the initial activation requires a significant amount of phonic gain. Thereafter, anybody can wield it at full power without needing to provide supplemental phonic gain in the form of song. 


Nehushtan was activated in episode 1 by the phonic gain from the Zwei Wing concert, and was thus usable by anyone (including Chris and Finé) thereafter.