> The risk of a false positive is nonzero, but how does it balance against the value of a true-negative?

Do you mean a miss?

Misses are unavoidable. If the perpetrator didn't have their phone on them, this is not going to help a bit. However invasive your investigative powers are, there will always remain a chance of misses. The question is how much privacy and freedom of innocents you want to sacrifice for a small further decrease of miss chance.