I you are alive, you already leave traces everywhere. No need for a needle or cotton swab here.
Of course I am being paranoid, but we all should be :)
It's true that DNA can also be 'non-obviously' collected. But there are two key differences between DNA and voice auth, when looked at from the perspective of a national surveillance agency:
1. 'Non-obvious' DNA collection occurs after the fact, and is much more costly. You have to send someone to the physical location, collect samples, and process them in a lab.
2. Because of (1), DNA could never be used as a 'always on' mass surveillance system. As opposed to voice auth, where you just need enough microphones, covering enough area, streaming data back to a server for processing. Sorta like how Batman finds the Joker in 'The Dark Knight' :)