I don't want to be pedantic but we're kind of glossing over some important history here. Alice Cooper, circa 1968-73, was the name of the band. The following year was their final LP before their breakup - Billion Dollar Babies. Two years after the breakup, Vince Furnier legally changed his name to Alice Cooper and released Welcome to My Nightmare. But to this day he still has to pay royalties to his ex-bandmates bc legally they own the name of the band.
This song is another one that was right in my junior high wheelhouse. We weren't terribly sophisticated - if it sounded good at 90-100dB, we were into it. Never really listened to him that much after BDB, didn't think he was doing anything unique and the stage theatrics were goofy (though I guess a few folks got their panties in a bunch.) That said, I like him as a person. He's engaging, loves to play golf (replaced alcoholism with chasing a little white ball), and even though he only lived in Pontiac for a few years, he's a loyal Detroit sports fan. Been on the Tigers and Pistons bandwagons since the early 70s. Seems like a good dude.