The thing is: it's not the White Council's job to extend trust, to look for more-charitable explanations, to hope for the best. They do not allow themselves to mind-probe motivations & intentions.
They go by actions and likely intentions.
Their job is to find and stop sorcerors before they do too much damage.
...
But Harry showed up to that party riding a 66-million-year-old necromantic monster. You don't get that sort of power in moments (I mean, Harry literally did... but so far as the Council is concerned, you can't... so obviously Harry didn't). What Harry did -- riding in on Sue -- proved to the council beyond all reasonable doubt that he had been secretly-studying Black Magic (to wit, Necromancy) for many years; in effect, showing that he was likely a disciple of one of Kemmler's disciples, and every bit as dangerous as any of Kemmler's disciples themselves!
Maybe, but they kept him on as a Warden for some time after that.. In fact wasn't that part of the paperwork that Rashid was going to fill out to get him reinstated in Cold Days? Can't remember if reinstating his Warden status was part of that or not.. Maybe he lost that when he became Winter Knight...
Well, if that's what they really think Harry is, then they blew that one BIG TIME! 
Harry's opponents on the Council would
agree with you there. That's pretty much exactly what they think. They think the Council has indulged a dangerous likely warlock for years, let him build up a serious power base, and opened the way to worse things. They mostly think Harry should have been executed years ago, when it would still have been easy and no bystanders need be harmed.
They would go on to say, though, that 'we are where we are', and for them, the longer the inevitable reckoning is delayed, the more painful it's going to be. So even if it would have better for most people to kill Harry 20 years ago, or 15, or 10...it'll only keep getting worse as time passes, the cost of the containment operation will keep rising, esp. for innocent bystanders.
Harry's allies take a more sympathetic view, and are more prepared to take a chance on him based on circumstances, but they can't
prove their position about Harry. From the collective Council POV, his enemies could still be right, based on the data.
I'm pretty sure there's also a swath of the Council that is not quite sure about Harry either way. (I suspect Carlos falls here.)
As a result, the Council position is perpetually muddied, and it has been for years. The ruling Carlos had to deliver looks pretty much like the result of a massive unsatisfactory compromise between the ones who want him dead yesterday, his allies, with the undecided voting a middle road. As with many (but not all) compromises, it satisfies no one and infuriates everybody.
Then, I'll say that the Council is riddled with villains, even among the highest members, because if they believe that Harry's actions are a likely long-term dark plot then they're basing it off what they would likely do in that situation.
That's pretty much standard Council procedure, and has been for centuries, for what it's worth. As Luccio observed back in the day, the Council isn't in the justice business, they're in the 'restraining power' business.
Why spare any kids that have broken any of the seven laws? Harry survived only because self-defense is permitted, and Eb took him under the Doom. Molly because Harry accepted the Doom and used politics on Langtry. Trust was extended.
And note that in both cases this was the exception, not the rule, and many Council members thought Harry should die at the time. Ditto Molly. In both cases, the result was an exception, and Langtry and his faction considered that it was a dangerous mistake in both cases.
Usually, a strongly-suspected warlock gets no second chance.
Why later promote a suspected warlock to the Wardens?
Because the situation was desperate, Luccio was one of Wizards somewhat sympathetic to Harry, and many people thought Harry being a Warden was a mistake at the time.
Why not move against Harry once he took over Demonreach? If you thought he was going to pull a Kemmler type power play, you left him all the unmonitored time he needed to set it up. For that matter, he could destroy the world at any time by throwing open the prison. Again, they trusted him not to do it.
You answer your own question.
Harry's enemies on the Council, I'm sure,
wanted to take him down once he became THE Warden. But they couldn't convince his allies to go along...and now he
is the Warden. So any attempt to take him down risks catastrophe if he threatens to release the monsters. So even if they're going to go after him, it's no longer simple or easy.
From the POV of Harry's enemies, his becoming Warden isn't a sign of trust, it's a sign that they were Right All Along and time is running out.
But he (technically)violated one of the seven laws during this pitched battle to save the world so he's now over the line and declared a convicted warlock that will be executed if he sneezes wrong. Why suspend the sentence?
Because the Council is divided and can't reach agreement. His enemies want him dead, his allies are his allies (to a point, anyway) and the middle of the road can't decide. So his enemies got a death sentence and his allies got it suspended. Compromise in a divided Council.
2
If he's truly over the line, they should have sent a strike team from Edinborough immediately.
And I have no doubt his hard-line opponents wanted to do just that. Strike hard and fast, while's he still tired and weak, and maybe you can take him out without him releasing the monsters or using his artifacts or getting help from Winter. But they couldn't get the rest of the Council to go along.
The Council's actions make perfect sense...when you remember that it's a giant committee, and even the Senior Council has to reach agreement among themselves.
