That's also not how they are used. They're maybe the username, but never the password, and absolutely not supposed to be secret. They are supposed to be extremely public.
I doubt the CPR number alone will give you access to obtain credit and the like today, but you can absolutely go into a pharmacy and buy someone’s prescription medicine with just their CPR number, and you for sure can get access to a lot of data by calling various entities and providing your CPR number as proof of identity (but at least fewer now than used to be the case in the past).
If the CPR number was truly made public information, those cases would be much more obviously wrong. The fact that CPR numbers are de facto considered pseudo-secret makes things much worse.
In Norway I have to show photographic ID and my fødselsnummer.
You for sure can't do this in Sweden. I very much doubt Denmark has gone total nutjob on this part, but if you have some proof they do please present it.
> If the CPR number was truly made public information,
In Sweden I can just call up the tax office and ask for the person number of anyone. A colleague was as confused about this as you are and he ended up intending to "prove me wrong" by doing it. They gave him my person number. Which of course they did. That's what it's for.
I am not sure what evidence you are looking for, but I have myself picked up prescription medicine using nothing but my CPR number on multiple occasions, including picking it up on behalf of my wife (providing just her CPR number and name).
They do so because they don't want to forget their password.