Yes, police are pretty good at solving "the spouse did it" crimes, the most obvious sort. When the victims were chosen randomly, the clearance rate becomes abysmal. They're also bad at solving crimes when the victims come from the marginalized fringes of society. Jack the Ripper is a famous unsolved case of a serial killer who targeted prostitutes more than a century ago. Modern examples include the Long Island killer, the Eastbound Strangler, and plenty more.
When the existence of such a serial killer is recognized, it tends to make the news at least regionally. Sometimes they become internationally famous for many years. But to my point, the public hearing about it is not contingent on the culprit being caught. If anything, the ones who are caught fast and easy tend to make the least amount of news. You can 'juice' unsolved crimes for stories a century after the fact, but stories that follow the "husband did it and we caught him" format tend to disappear from the news after the culprit has been sentenced.