Some arguments I'm seeing:
* "Privacy": I don't see this more private than the information contained on a driver's license. It also does not expose anything that's not otherwise already available to the police. I don't see the difference (when it comes to privacy) between DNA and a fingerprint that you have to provide when getting ID/drivers license.
* "False positives": Yes, they'll probably happen. But even if we go with that assumption there are additional ways to exclude most of them. And those that can't be excluded get tested for an exact match and/or additional evidence. This is no different than being a suspect in any other way.
* "False convictions": As far as I understood it this will just be used to narrow down potential suspects, not to get a conviction.
* "Police will lie do anything to convict the person just based off of this evidence": That's a separate topic that needs to be addressed, but not by reducing tools and forbidding technology that can narrow down and find suspects.
Under worst case scenario is being a false positive result and maybe talking to the police (if you were in the same city, your age was correct, you were around/close to area of the crime) and providing exact DNA sample so much worse than the fact that "The approach has led to arrests in at least 60 cold cases around the country."? I say it's not.