Covering up the cure for Tranquility is only a tertiary aspect of the Seekers' corruption. It was, primarily, the greater neglect of their given duties, to such an extent that they actually helped to upend the system they were supposed to preserve.
What corruption of what duties? This is not an idle question- this is about system practices, not the views of individuals, and what they actually are reasonably expected to do. 'Act the way I like' is not a meaningful standard.
'Corruption' as an organizational concept has two meanings. The dominant meaning, relevant in modern political discourse, is 'dishonest or fraudulent conduct by those in power, typically involving bribery.'
There's precious little to suggest that there was systemic bribery within the Seekers, even fraudulent. Of all the Seekers we've met, only one had his services 'purchased' by an outside offer, and that was Lucian- and even that wasn't for money. Meanwhile, Lambert was a hardliner, but not a fraudulent one- he believed in a hard stance against the mages, and was looking for a justification for it, but there was reason. He didn't invent justifications or frame the innocent for his suspicions- he was right about there being a murderer, and there was a independence movement, and Justinia was sabotaging him and the Templars. None of these were 'bought' views- just as there's nothing to indicate the Seeker investigation of Kirkwall was fradulent. Unsatisfying? Sure- you can place more blame on whoever you want, and not agree with the conclusions, but the reasonings for the Seekers were not baseless. Kirkwall was a problem, and it wasn't just Meredith. In so much that the Seekers put pragmaticism above purity, this is critiquable- but it's also the nature of actually having and exercising power, rather than being a faltless observer.
The Seeker investigations of Kirkwall seem to be the greatest 'failure' of the seekers corruption, so I'll hit on it here.
Even if you posit that the 'proper' answer should have been removing Meredith- if there's any good point to remove Meredith that doesn't rely on meta-knowledge and wouldn't threaten to make the situation worse- it doesn't change the fact that for all her paranoia, she was right about real, legitimate corruption in the Kirkwall Circle. The First Enchanter was a blood mage protecting for a serial killer out of political convenience. Maleficar haunted the streets. Saying that it would have been better even so, so long as the tensions died down, isn't an argument about purity from corruption- it's an argument that corruption is preferable so long as it keeps the peace.
And honestly- cracking down on Templar corruption was one of the problems with Meredith inflaming tensions in context of Kirkwall. Even what's-his-nuts, the tranquil rapist, had to resort to secret crimes- you know, the sort that the Seekers wouldn't have naturally overturned if they were investigating Meredith- and this was when Meredith was a hardliner. In the first two acts we get multiple references that Meredith is cracking down on corruption- Hawke needs big money because bribes to avoid the Templars are getting harder, fewer Templars are willing to take them, and those that do (like Samson) are getting pushed out. Meredith cracks down on the mage underground, and the lyrium smugglers, and the likes of Anders mage-freedom underground. In the first two acts, through the Qunari invasion, any fair Seeker investigation that can look to what we know wouldn't have found Meredith to be the root of corruption- it would have found her to be trying to root it out rather than the comfortable entrenchment that had existed before.
It's only come Act 3 that Meredith starts stepping outside her bounds for reals- but it's also here that removing Meredith would do the least 'good' in the name of countering corruption, and could cause the least immediate harm. By the time Meredith becomes insane, she's not only the most poltically powerful person in the city- and thus trying to remove her risks a costly fight- but she also has a real and immediate menace to face with the maleficar movement, Tevinter agitators, and so on. Reigning Meredith in and stopping the Templars doesn't solve a blood-mage first enchanter, or the maleficar in the streets. Continuing the crackdown under new leadership doesn't change the tensions. Either corruption continues by empowering Meredith's enemies by taking her down, or by tolerating Meredith's abuses- but unless you can take down both sides (and there's precious little to suggest that's possible without remarkable cost), there's no corruption-free approach. I honestly don't see what a Seeker investigation could conclude that would 'solve' Kirkwall in a corruption-free manner.
That's all the direct form of corruption- the one most people refer or insinuate when calling an organization corrupt. But there's the other meaning as well- "the process by which something, typically a word or expression, is changed from its original use or meaning."
This is the other use- and can be used on the Seekers- but not as simply as many suggest. Yes, the Seekers (and Templars) have changed since their intial formation- and in most cases that's considered an evolution, a good thing. It's also been incrimental- neither Lambert or anyone else woke up off the bed one morning and said 'hey, let's do something different.' What matters in the context of the Seekers is 'what is their purpose?'- and how could they have avoided being corrupt.
There's two frames of reference here when it comes to this usage of corruption. 'Original purpose,' in which the original form of the thing is the proper/correct form, and 'traditional purpose,' in which even as things change over time the norms of generations do change, but this is an acceptable and good thing and the current norms are the 'baseline' from which corruption is measured.
In either case, it's hard to claim the Seekers were corrupt- or at least, anywhere near as corrupt as Divine Justinia.
For original purpose, the cornerstone we'd have to look at is the Nevarra Accords- the founding agreement that founded the modern chantry and its relationships with the Templars and Seekers. It's occasionally said that the Circles were never meant to be prisons, but rather refuges. Arguable- the distinction between a prison and a refuge is often state of mind-but a far more radical deviation from the origin is the actual disolusion of the Circles entirely. If the Seekers did NOT leave Justinia because of her intent to allow the Rebellion to succede as fait accompli, then they really would be complicit violating the original purpose of the accord. Which was, ultimately, incompatible with mage independence and integration, which are the goals of the mage rebellion. Under founding purpose corruption, leaving the Chantry is the only anti-corrupt path so long as the Chantry intends to allow the mages to leave. The legitimacy of the arrangement between Chantry and the Templars is based on the agreement, not the Divine herself- if the Divine violates the accord, she is corrupt. The original purpose of the Seekers and the Templars is not to be her lapdogs and do whatever she says because she says it.
Now, original purpose does have one fair criticism for the seekers- that they should have been more impartial between mages and templars. This is true- but not particularly relevant, since most of the contexts we see are not systemic sorts where 'unbiased' would have necessarily done anything different. As was argued above, even an 'impartial' Kirkwall investigation would have had a hard time doing differently early on, or avoided implicit corruption come Act 3. Lambert was certainly biased, but many of his actions were against corruption or actions that even a 'fair' Lord Seeker couldn't have countenanced without tolerating corruption. And once the rebellion again, the measure of closer original purpose is probably going to fall with 'meet the intent of Nevarra accord in maintaining Circles' than 'allow collapse of Nevarra accord in name of Divine.'
'Traditional purpose' is the line of thought being implied when corruption is cited by people who are thinking of Lambert's opposition to the Divine. This appeals to more current norms: that because the Seekers have become the personal agents of the Divine, that because the Templars have spent a millenia following the Chantry's orders, they should continue doing so regardless of the Leader and her views. The Seekers and Templars legitimacy comes from the Divine, not the Neverra Accord. Obey the orders you disagree with because that's the purpose, and corruption is defying that purpose.
The thing is, though, traditional purpose works both way- and Divine Justinia can still be called, with some basis, corrupt. Yes, it's true that the Seekers and Templars answer to the Divine by tradition- but it's also true that the Divine has traditionally upheld the Circles and not allowed their dissolution. Even without the Nevarra Accord, Justinia's decision to let the mages revolt- indeed, to enable them to do so- is a corruption of the traditional purpose and traditional relationship. Traditional purpose legitimacy doesn't just bind the lower subordinates- it binds the upper ranks as well, and when the top is corrupt the traditional legitimacy doesn't simply demand obediance- obedience is within the wider context of what the obedience is for, whatever the current norms are. Justinia might have been allow new norms to develop- Leliana certainly tries to create new norms- but at the time of the Seeker and Templar rebellion, there's no tradition for Justinia to appeal to.
The other bad thing about traditional purpose- at least as far as calling the Seekers corrupt goes- is that the original purpose doesn't really matter so long as the current reasons and traditions are reasoned. Tranquility is an established part of the Circles, legitimized by the mages themselves- enough so that, if Tevinter is a guide, they won't give it up if they could. Tranquility as a punishment is not a norm- not yet, and fully deserves to be fought back against- but the Tranquil Cure doesn't real resolve anything- the vast majority of Tranquil are Tranquil because mages wanted them to be. The Tranquil Cure is a political firecracker, but doesn't do much to make things better- it's not really an issue of purpose-corruption under traditional purpose. Similarly, even if the initial intent of the Circles as an anti-integration system is less relevant, the current tradition very much is- the traditional purpose of the Circles is Mage-Mundane separation, with Templars keeping it.
Mage independence and dissolution of the Circles is corruption under the traditional intent. It's also corruption under original intent. Which is perfectly fine- as long as you accept that 'corruption' doesn't inherently mean 'bad'- but it does make an awkward argument against the Seekers by any pro-mage independence perspective. A less corrupt Seeker organization should have taken a more even hand towards the mages- but ultimately, the only way the Seekers would have supported mage independence in any way, whether by action or inaction, would have been... corruption.