Magnitude of a crime can always be expanded to justify anything. We have plenty of things we could do to catch more criminals (even murderers!), but we don't. As an example, why not run a DNA check on any blood draws at the doctor's office? Sure it's an invasion of privacy, but what if it catches a murderer!
I get it. Here the argument is that the person approached law enforcement on their own, and thus opened themselves up to this. But this is incidental -- it happens to be that reporting this crime requires handing over DNA. But police could require everyone to do a blood draw when reporting any crime, right? I think most people would have a problem with that and think it is obviously an invasion of privacy. But OK, you can argue that the real problem there is that it is too tit-for-tat. How about instead every time you walk into a police station someone watches you and collects any hair that falls off of your head and runs it through the database? No coercion, no one can argue that they had to give their DNA in exchange for services. Still seems like it would still have a pretty negative effect on trust between people and police.
The only thing that makes this case "confusing" is that a bunch of different actors are bundled together. If the administration of a rape kit was 100% done by a doctor (as far as I know it is done by nurses but then handed over to police, currently?), and they could then trivially separate out the two genetic materials and hand over only that which didn't belong to the victim, then I think most people would find it distasteful for the police to demand both sets of DNA, right? Why do you need the other one (the doctor could confirm it belongs to the victim if that is the concern)? Just because it happens to be that the way we happen to administer this today makes it convenient to check both (and perhaps even easier than not checking both), does not in my mind introduce any sort of new ethical consideration. It still feels an awful lot like "well, if you want us to look at your crime, you have to be willing to have us check you for any crimes first".
The job of police is difficult, not as an accident of history because they didn't have the right technology when we wrote a lot of these laws, but because it's supposed to be difficult because we understand the value of what's lost if it isn't.