Actually, if you look at early footage of them, you can see elements of what would become "glam" already in their outfits. I surely recall the WigWhambam era, and the facepaint and satin gear, but I'm talking about before it as well.
As for their saying they were taking the piss, you know that most, if not all, of those bands like to deny they ever were glam, or else they use the piss angle. Sweet protested too much, in my not so humble opinion. Andy, especially, wanted so much to be taken seriously by the likes of Rolling Stone, him being a prog(Elastic Band)and all that. They all were really, but him especially. Recall the media despised glam, I'm sure you do.
As for Buddy Rich, yes, the influence of those ornate drummers from the Big Band Era was felt a lot in glam. Mick got his chance in Man With The Golden Arm to show his stuff.
BTW...that about Cozy is on a site about him.
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