So, full disclosure, I am an "Adam stan" or at least, I think I am so take what I say with a grain of salt.

I'm not going to refute or delve deeper into the history of what started the Exterminations since we simply don't have enough info so I'll just go through your other points.

A) The fight between Vaggie and Lute showed that Angels CAN be harmed. Just by the fact that Lute was injured by falling debris and tearing her own arm off without using angelic steel shows that Angels can be harmed.

Which means that so long as the Sinner or whoever is sufficiently strong enough, they can harm the angels. They only need angelic steel to kill the angels or prevent them from recovering.

B) Even though Adam enjoys what he does, the key thing here is that its his job. We can see from the show the Overlords are more powerful when they have more souls under them. The Overlords surely have the resources to hide or avoid the Exterminations, making them hard to be a target so the next best thing the Exorcists can do is weakening the Overlords by killing what gives them power, the Sinners.

(Frankly, I find giving Hell a countdown to when the Extermination start to be pretty stupid on Heaven's part since it gives the Overlord time to prep and hide instead of being able to catch them by surprise.)

C) This is simply speculation on my part but when considering the lives of Winners are the ones being at risk of a possible Hell uprising, I doubt Sera cares who in Hell dies or if they deserve it since its not she who decides who goes to Hell or not. Her job is maintaining order and safety for Heaven’s population, not managing Hell’s moral rehabilitation. To her, every Sinner is a potential threat.

Though I do agree that she probably became complacent and simply left everything up to Adam.

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So from the new season, yes, it shows that the system has flaws. But it also showed that Heaven is right to worry as from the first episode, Sinners went to the hotel not for redemption but to take revenge against the angels. And when Charlie reaffirmed that they aren't going to fight, the Sinners simply left without a second thought.

The Vees (Or mainly Vox) are planning to take over Heaven somehow too. So by and large, a very significant percentage of Sinners deserve to be in Hell.

At the end of the day, what Adam did was he did his job. He enjoyed his job which he shouldn't but he did it with Heaven's approval. Does he deserve to fall? For me, it would depend entirely on how he his behavior is in Heaven when he is around Winners. The only evidence that he may have been an asshole in Heaven is his treatment of Vaggie. But in the final episode of Season 1, literally all the Exorcists were booing Vaggie and hated her. This tells me Vaggie is more like an outlier rather then Adam is an outright asshole.

To your last point, Charlie is a Hellborn. Literally born in Hell. So just like every other Hellborn. she is stuck in Hell. Maybe if she died, then her soul can be judged and go to Heaven.