Why the hell are people surprised ASO gave Puy de Dôme HC status? They've kind of started giving out some rather tame HCs in recent years (as mentioned, things like Arcalis, Plateau des Glières, Pla d'Adet, Finhaut-Emosson and Col de la Biche have been given that status recently), and Puy de Dôme is both tougher and more historic than those.
Yes, the last few kilometres' steepness means that's where all the action is, but that's no different to Col de la Loze, just that that's longer. As mentioned by Netserk, there's similarities in the profile of Puy de Dôme to the Passo Fedaia in shape and profile, and in characteristics it's also similar to Bola del Mundo only if the hormigón section was paved and was 50% longer. I don't see it as too dissimilar to the likes of the Col de la Joux-Plane which is one of the other borderline HCs. Definitely I feel ASO goes on feel (and likes to over-categorise climbs with altitude as an additional factor), as there are other climbs like Col de la Ramaz and Col Luitel which have never been HC but are tougher than many which have, even leaving out the obvious outlier of La-Ruchère-en-Chartreuse.
Puy de Dôme is most definitely an HC climb by the standards of modern ASO.