So for the Bourse, the answer is that its functioning is a lot more machinelike than the other organs of state. Money talks, and there are very few things it can talk about. A foundation of the Empire IC is property rights, and a foundation of the game OOC is the unfairness inherent in a market economy. What would the Synod do - reverse the outcome of an auction it didn’t like? OOC it all relies on the in-character money actually being worth something, and that means that the full faith and credit of the state needs to be maintained. So the Synod can’t have too much power to mess up the Bourse or people will abandon the ring in favour of your usual LARP barter.
For Conclave - and I say this as an avid Conclave goer in my day - Conclave is very information-dense. It’s the intersection of roughly speaking six specialist subjects with different modes of operation, reacting to an environment that can change much more rapidly than the religious zeitgeist. You can raise something a minute before the motion deadline, canvass people in the queue to get in, speak well on the floor and have new law by the time it’s finished - and you may need to. The Synod necessarily slows everything down. To get a veto through the Synod is a solid day of hard legwork by you and as much of your network as you can muster. OOC, the powers aren’t a good fit for each other.
The other point is that unlike most of the other arenas of Imperial power, it is somewhere that you can turn with a good speech. You can get a very great deal through Conclave if you can bamboozle a dozen opinion formers, fake a pile of specialist expertise and ride herd on a crowd of a couple of hundred interested observers - Conclave makes mistakes, Conclave can be fooled, and that is part of its charm. Out of character, the Conclave game would be easier and in my opinion less good if bad/hasty/fast-talked decisions enacted at 10pm Friday could be reversed before 9pm Saturday. I think the whole of that part of the game would be distinctly wounded by providing a Backsies option.
In character, of course, magic is a deeply complicated subject, and as mentioned the Order of the Sevenfold Path is absolutely there to make sure that Virtue is considered in magic. And they are right there, part of the action, so they can act fast and call things out as they see them. Part of my old character’s advice to anyone trying to make it in Conclave was to get religion, or pretend to - it’s a major tool in the arsenal of any archmage.