Can't keep DNA secret anyways, we shed it constantly
Our DNA sequences are impossible to keep secret from government (or anyone else) anyways. Our DNA is on the clothes we wear, the cups we use, and the paper products we discard.
The problem arises because forensic DNA matching does not match the entirety of the DNA, it is really a spot check of the size of various segments. There can be false positive matches.
The testing companies say the odds of a false positive are 1 in 1,000,000,000 to something like 1 in 1,000,000,000,000,000. For example, identical twins would totally match. But also, we have isolated villages in northern Canada where populations of a few hundred people are descended for 3 or 4 people who created the community 200 years ago. I can imagine there being false positives there. (Red Sucker Lake, Manitoba is an example.)
The problem is, do DNA analysis matching techniques sometimes cause false positives, where the wrong person is identified. With databases where only a portion of the population is recorded, this could happen and nobody would know.
It would be much safer for all to have a universal database of all residents and visitors to be in the database.
That way, if there are (say) 7 people in the UK with DNA that matches, all 7 will show up, rather than just the one for the person with the previous conviction. The police will realize they need to do more investigation, rather than jumping to conclusions.
That way any frailities of the technique, false positives, etc., will quickly become obvious.