Why did the RIAA &MPAA target peer-to-peer networks instead of one-click hosts?

A few years ago, the RIAA (and to a lesser extent, the MPAA) when after uploaders on peer-to-peer networks like Limewire, thus diminishing the popularity of those sites and leading to some people settling for $3,000 or so for downloading music. I think the main thing it did was made many users downloaders only, which killed selection choices.

Over the weekend, I read an article in the New Yorker where a porn company has filed about 3 lawsuits a day by hiring consultants to trace uploaders on BitTorrent LINK

Apparently this one company is now the source of 1/3 of all copyright lawsuits in the US, as the RIAA isn't suing anymore.

So here's my question. Why do the parties suing individuals always go after peer-to-peer users rather than users on single click hosting and/or streaming sites? Is it easier? Does it require fewer subpoenas to access records? Or do IP addresses disappear when uploaded to a storing site like Rapidshare or one of the streaming sites? Aren't they basically the exact same thing otherwise?

Doesn't make any sense to me. Unless I'm missing something, wouldn't everyone just switch to other sharing outlets if they know one kind is being targeted by professional organizations and/or copyright trolls?