Except that it's not propaganda, it's literally what we see from the inside, from someone of high rank in the Colleges who works with a lot of people from lots of Colleges, including those of high rank.
You also contradict yourself: Grey Magisters who horribly abuse their power get murdered by the Grey Order. While the truth of the matter might stay classified, the fact of the matter is that the Grey Order considers the discovery, tracking, and elimination of Black Magisters to be one of its most important duties.
The Grey Order also goes to significant lengths to keep track of its wizards and keep tabs on their general activities. Murders leave a trail, and when those murders line up with specific wizards' presence, investigations ensue. We see this kind of oversight with the dark rituals--they keep close track of who knows what, and when, and if anything like the dark rituals that are shared are employed in the future, they will need to know where the people that were taught those rituals were at the time.
And you're also really twisting events to look at things in the most negative way possible. Regimand murdered a proven traitor and vampiric conspiracist after the Witch Hunters had gathered ironclad proof of her guilt. The only reason the Colleges performed the deed was because the head religious and legal authorities asked the Grey Order to do it instead. Similarly, with the late-Tzar, Mathilde was asked to by the Prince specifically because the world was at stake and because he would launch a civil war or coup if she didn't do it, and she'd seen first-hand how badly the late-Tzar's rule was impacting Kislev from a variety of perspectives (and first-hand).
She didn't kill him because he wasn't good at his job. She killed him because he was in a position of enormous responsibility and he was very stubbornly, negligently, and complacently refusing to do his job, and all of the alternatives were far worse for everyone--if she didn't do it, far more people would die for a fact. Even then, the act is something that they Grey Order would not and cannot sanction. Mathilde will take that secret to her grave and should it ever come out, she will fully be expected to take all the blame and die for it. The Grey Order does not willfully turn a blind eye to this sort of thing, so only those who really prove their character are taught the spells necessary to be that kind of potentially exceptional assassin.
The motto of the Grey Order is very much "do the right thing, or at least the least bad thing when necessary". "Don't get caught" is not their motto, because as Mathilde's master taught her, for Grey Wizards doing what they do best, there very well may come a time when slitting throats of the enemies of the Empire results in you having to sell your life at the highest cost possible to the enemy. Sometimes getting caught happens, and so every throat you slit is a cost-benefit calculation. Are you selling your life at a high enough price?
Mathilde has assassinated a shitload of greenskins, skaven, Chaos cultists, and necromancers. She has assassinated one noble and one member of royalty, the first of whom was thoroughly proven to be guilty of capital crimes against the Empire (and the second of whom I already went into above). Assassination is reserved for capital crimes; otherwise, you get stuff like "sneak into their home and get evidence of potential crimes, and we'll see how serious it is" or "infiltrate this group and see if they're up to no good". Since this is Warhammer Fantasy, there are a lot more kinds of capital crimes than there could be IRL.
Do you remember the Night of a Thousand Arcane Duels? That time when all the wizards decided to have a battle royale in the middle of Altdorf?
Where were all of those noble, honorable College leaders then? I also notice that you didn't mention Alric. Truly a paragon of wizardly virtue! A man I absolutely trust to distinguish between what is good for him and what is good for the Empire.
Grey Magisters who horribly abuse their power get murdered by the Grey Order
when they get caught. Or, to put it another way, Grey Magisters who embarrass the Grey Order publicly can expect to die. Institutionally, the Grey Order places great emphasis on the
appearance of being incorruptible. The lesson here is simple: Don't get caught.
The Grey Order goes to significant lengths to keep track of its wizards? Wizards who are trained in disguise and deception? Wizards who are, by virtue of their education, exceptionally good at not being tracked?
Establishing a fake cover story is, like, baby apprentice stuff. An advanced journeyman could probably drop plenty of corpses without any sign that they were within a hundred miles of the dead bodies, and that's not getting into what a full magister could do. The problem with training your people to be really sneaky and good at not getting caught is that you're training your people to be really sneaky and good at not getting caught.
I'm pretty sure that the Emperor is the primary legal authority the Colleges answer to. It says so right in the Articles of Imperial Magic. I wonder how he'd feel about them deciding to kill his wife without telling him? I guess we'll never know, because they didn't tell him. Why bother to say the Emperor is the boss at all, when the Grey Order will just do whatever they want and say that their actions are in keeping with "the ideals and laws of Sigmar's Holy Empire"?
The Tsar wasn't good at his job. You can say that he was stubbornly refusing to do his job, but I would disagree. Vladimir was trying very hard to do his job. He was a good general, a fierce warrior, and a very devoted man. It's just that none of those qualities made him a good head of state. He didn't
want to fail as a Tsar, he just sincerely disagreed with his son.
The Grey Order doesn't seem to "sanction" anything. But Mathilde was raised in the College, and she feels no moral expectation to refrain from killing an allied head of state. Just as there was no moral expectation for Regimand to refrain from murdering
his boss's wife. Grey Magisters feel that they have the right to serve as judge, jury, and executioner, regardless of rank. The only rule is that if you get caught, you're on the hook.
The list itself wasn't the proof; it was the end result of investigations and proof-gathering.
Totally no way that the Lahmians, who specialize in infiltration and deception, could have planted evidence against the Empress. Absolutely not.
Remember, the Emperor is legally our boss. Second Article of Imperial Magic! The Grey Order doesn't care, though, so Regimand decided to murder his wife without telling him.
I feel that this was a fairly dubious decision, because if the Emperor found out he could
absolutely go after the Grey Order. However, the basic rule of the Grey Order is "don't get caught", and Regimand didn't get caught, so he's golden. Articles of Imperial Magic? Eh, Magnus intended for
us to interpret what the "ideals and laws of Sigmar's Holy Empire" are. Who needs to consult with Sigmar's Holy Emperor before you assassinate his wife?
Edit:
I can't see in the update much discussion about proof gathering or in-depth investigation.
I think it's likely that Regimand had to make a very difficult judgement call on the basis of inevitably inadequate information and made the decision that the risk of leaving a potentially compromised Empress in position to influence her husband and soon to be born heir was worse than the consequences of assassinating her.
Arguably, that's his (unofficial) job as the Grey Magister on the ground.
I feel that there are sound arguments against having an Empress who is likely to be compromised by Lahmians.
I also feel that the Emperor and the Elector Counts would
absolutely say that it isn't his job to make that call. If the Grey College stood up and openly said "The Articles of Imperial Magic allow us to murder the Empress without trial", they would get stabbed. The wizard secret police is completely off the leash, with zero accountability to our legal boss, but we got away with it.
This time.