This is not a terribly popular opinion online, but: the cops don't care about violent online threats, because the vast majority of them are uncredible.
Virtually every public figure gets violent online threats. I've gotten them, and my closest claim to fame was "I was in the top 5% of Reddit comment karma at one point". This just happens. It is a really really really shitty social norm, but that's what it is. Police are unwilling to act on it, because in the majority of cases, they are unable to act on it. Attempting to respond to uncredible violent online threats would only waste their time and our money, and prevent them from responding to more actionable problems.
Of course, the discussion of 'violent online threats' isn't the only thing in this article. She's also talking about 4chan harassment. That's not uncredible violent threats, that is illegal harassment. That should be taken seriously, although again it may not be actionable. I confess, I'm somewhat allergic to Gawker and didn't get through this article, so I'm not sure if the harassment is what she means by "violent threats". If it is, and police aren't taking that seriously, then that is a problem. I would be seriously surprised if they don't take that seriously, though