Now it does make sense. Thanks to both of you!
One thing I did get clearly was just how Sakura was "controlled". In point of fact, the emotions she felt as Dark Sakura were entirely her own. Thanks to Matou Zouken's worms, her libido was driven through the roof - creating an ever-increasing lust for Shirou, and consequently an ever-increasing envy of Saber, Rin, and Ilya, who she saw as "competition that Sakura was losing ground to". Further, Zouken taught Shinji to heap abuse on Sakura (strongly hinted at, and possibly confirmed at some point I missed, that Shinji repeatedly raped her) - there are all kinds of issues that spring up from that.
Her connection to Angra Mainyu came from the worms Zouken put in her. Specifically, Zouken had taken broken shards of the Grail from the Fourth Grail War and merged them with some of his worms, which he put into Sakura. This effectively made Sakura a candidate for being the new "body" of the Grail (supposed to be Ilya's job), and when Shinji tried to blackmail her, she snapped and her dormant magic power joined with the tainted Greater Grail - effectively making her and Angra Mainyu into one, giving her access to endless magic power (from the limitless potential for evil in mankind?), and making her very scary.
It certainly wasn't mind-control - IIRC, she changed back pretty quickly when she decided she wanted to.
Hmm. I remember reading that, though there is only one Greater Grail, there is a new Lesser Grail for each Heaven's Feel ritual. Could that mean that Avenger was the Lesser Grail for the Third Grail War?