Part of the problem is that as the size of the database grows -- i.e., as every police encounter turns into a mandatory sample collection -- the false-positive rate will also go up.

Remember: DNA "matches" are probabilistic, and most of their utility has been based on the low probability of a sample taken from a crime scene "matching" a sample taken from a suspect. Prosecutors (committing the prosecutor's fallacy, of course) love to tout the "one in a billion" type statistics for a match being a false positive, but it doesn't take a very large database at all for "one in a billion" coincidences to start occurring.