You've unwittingly demonstrated why it's impossible to have a "trusted government", and why governments are inherently awful even without nefarious motives!
> "we found DNA on this woman's clothes and immediately matched it to this man who hasn't committed a crime before"
You've casually thrown out a use that sounds like it would be valuable, and thus has political appeal - but it actually fails hard to false positives. If you're merely confirming a DNA match, then the small chance of a false positive remains small - you're only comparing against one candidate. But if you do a population-wide search, then that chance of finding false positive multiplies [0] across the whole population! So even with good intentions, you've invented a process that picks random people out of society and harasses them (at best, with the way justice systems actually work this would lead to false convictions).
> "we found DNA on this woman's clothes and immediately matched it to this man who hasn't committed a crime before"
You've casually thrown out a use that sounds like it would be valuable, and thus has political appeal - but it actually fails hard to false positives. If you're merely confirming a DNA match, then the small chance of a false positive remains small - you're only comparing against one candidate. But if you do a population-wide search, then that chance of finding false positive multiplies [0] across the whole population! So even with good intentions, you've invented a process that picks random people out of society and harasses them (at best, with the way justice systems actually work this would lead to false convictions).