You already know what you really need to know.
Yes, the original Alice Cooper band's material was in fact written by and large by various members of the band. It was an incredible creative force in both songwriting ability as well as instrumental acumen and talent.
Sadly, the band never truly broke up as much as it ceased to exist. Glen Buxton ( lead guitar player) delved into drugs (a Vincent Furnier no-no), Bob Ezrin (the producer) had already split after "Billion Dollar Babies". "Muscle of Love" was not anywhere near as big a "hit" as the previous records (although I thought it was still very good).
Alice thought more and more about going solo, and people got into his head (Dick Wagner, anybody - a talented guy but less talented in my eyes than AC or his former band members). Next thing you knew "Welcome To My Nightmare" was out and it began
my nightmare, because although it was not garbage, it was
definitely a sell out to commercialism.
If you'd like more, I recommend the following...
http://www.sickthingsuk.co.uk/content.php?id=misc%2Fbreakup.php