Aside from the huge privacy issues, I'm even more concerned that they would screw up collecting it. Everyone sheds DNA. Who is to say that the DNA they collect is actually from the person they think it is? DNA gets all mixed together after it leaves a person's body.
That's what juries are for. The prosecution presents evidence that the defendant's DNA was found at the scene of the crime. The defense attorney then attacks the credibility of the evidence, for example by offering an alternative theory for why his DNA was there, or attacking the method of collection to raise doubt that it's actually his DNA the police collected. (He doesn't necessarily have to provide his own DNA sample. Defendants aren't obligated to prove their innocence.) Then the investigator explains why the collection method is sound. At the end of the day, the jury decides whether the defense has raised a reasonable doubt.