To do that it sounds like they may advertise themselves on the dark web as an attack web site. They also don't bother to verify ownership of a target, like checking domain ownership for example is probably the most common use case here. They also use other people's mis-configured devices to amplify their attack. That last part is particularly salient I think.
It's very stupid to advertise yourself on the dark web for this if you host a site on the clear web. But you are probably right that LE lets this goes on for longer than it needs to. I don't know how long they have known about these sites or how they go about finding them. It sounds like it's just based on victim's sending in reports.