The criticism I had heard of Billy Idol from the old guard was that he wasn't "real punk rock."
With Generation X, he was marketed as a teen heartthrob. He also enjoyed session musicians and people writing hits for him at one point. I always dismissed that criticism because he was just the next wave of punk, younger than the Sex Pistols first wave and therefore rose to his peak during New Wave.
I also appreciate the serendipity of teenage Billy Idol appearing on Marc Bolan's final episode of "Marc". He passed the torch before his fatal crash. Luckily Billy Idol survived his accident.
As a member of the 'Bromley Contingent' and a close friend of people like Siouxsie Sioux and Steve Severin, Idol was there from ground zero and was partially responsible for the large following that Sex Pistols (briefly) enjoyed. You're right that Generation X was among punk's "next wave", and their music tended to not only be more optimistic in nature compared to the likes of the Pistols, but they also wore their classic rock influences on their sleeves when it was popular to dump on those who came before them as quaint and old-fashioned.
But he had a whole other career beyond England's punk scene after coming to the US. By embracing the bourgeoning New Wave and pop scenes and combining his classic rock influences with his punk looks, not to mention taking full advantage of the nascent MTV, he created truly great rock and roll with an edge that is as iconic and emblematic of the 80s as any other artist you can name from that period. And some of his stuff, especially
Rebel Yell, has gotten better with time (inconceivably, that song peaked at #46 in the US, but has become an anthemic powerhouse of that decade through regular radio play.) And not only did he survive his motorcycle crash (which prevented James Cameron from casting him as the T-1000 in
T2), he won his battle against drug addiction, tours regularly (saw him last year in Cincinnati, he kicked ass) and has challenged himself musically with a couple of acclaimed albums since the early 2000s (him and Stevens just wrapped up production on a third.) Finally, he had to be shot with a tranquilizer gun from the army after destroying a hotel in a drug-fueled, hooker-filled, three-week binge in Thailand.
The guy is the living embodiment of a rock star and his absence from the Hall is a travesty.