1. They do not know the intelligence is on the Citadel. It could be any were in some remote system or buried deep inside a planet or even be an actual planet for all they know.
2. The second the Reapers learn the Crucible needs the Citadel regardless of them knowing about the AI hidden in it they quickly attack and take over the Citadel with ease. Transporting it to Earth protected by the largest Reaper Fleet assembled in one spot. While still maintaining enough Reaper presence across the galaxy to continue with the harvest unobstructed. This fleet is still shown to be capable of engaging and wiping out the ENTIRE FLEET OF ALL RACES OF THE GALAXY.
3. If said intelligence was attacked by any of the races attempting to destroy it to effect the Reapers it would simply summon any and all near by Reapers to defend it from harm. Resulting in similar set ups that have already echoed across the galaxy. Organics put up a fight but are ultimately destroyed.
4. During the entire Reaper invasion 1 of 2 things are happening with any system the Reapers invade. 1 the population evacuates who ever they can leaving the rest to buy time for the others to escape and being left to be harvested. Or 2 the race(s) slow the Reaper advance at the cost of unsustainable casualties.
1) I already said that. That's not a reason to not try to learn about it.
2) And yet somehow they can't stop the Crucible from docking with the Citadel or stop Shepard and the Crucible from Destroying all of them.
3) And how long would it take for the Reapers to get there and stop them? Maybe they'd have some sort of suicide mission. Maybe it'd be too late. There are ways to set this type of thing up. It's been done before.
4) I don't know what your point is here. Secret projects continue to run.
What you seem to be missing is that the characters don't even have to do anything big. If they actually talked about it and gave any of the excuses you did as to why they can't actually act on it, that would be better.
I make no straw man you are simply more interested in being right rather then analyzing what is happening in the game world. You want to see them being stupid by not going after this mysterious guiding intelligence that no one has any clue exists or any idea of were to find it. Even Leviathan wouldn't have any idea were to start looking. So to satisfy that set up you are ignoring everything the game has shown is happening because you want to be right rather then correct.
The Reapers have the races of the galaxy against the ropes. It does not matter that there is some intelligence guiding the Reaper's actions the fact is the Reapers are actively attacking hundreds of worlds causing billions of causalities as they try and defend them. They need to take care of the Reapers first. If not out right destroying them all but simply getting hold of a weapon that allows the races of the galaxy to create a safe area that they can hold out against them. This links to my D Day example. The landing on the beaches of Normandy existed solely to give the Allies a secure foot hold in Europe so they could advance to the interior of it and ultimately Berlin.
Once the Reapers are either out right destroyed they could devote their time to finding that intelligence. Or if they at least gained a weapon capable of taking them down it would allow them to rally around it and at least create a defensive area that they could defend and give them time to find it.
It litearlly boils down to 2 choices
Choice A: They go with the long shot Crucible item that they physically have in their hands partially build and hope it works as it seems to claim to
Or
Choice B: Abandon the only tangible long shot they have to waste time chasing down rumor and speculation that only the Reapers would truly know anything about. All while the Reapers continue to harvest planet after planet.
You obviously don't know what a strawman actually is since you do it right after saying you don't.
I never suggested choice B. I didn't suggest abandoning the Crucible, but to make it the only thing they pursue is equally foolish given how little they know about it. There is no indication that they have to put literally all resources into the Crucible with nothing left to spare.
In LotR, Frodo and Sam went alone into Mordor while others saved Rohan and Gondor. After the battle of Minas Tirith, they attacked Gondor to give Frodo cover.
In The Matrix Revolutions, Neo goes into the Machine City and negotiates with some Central Intelligence while Zion is under attack.
Even in the terrible Phantom Menace they attacked the droid control ship with fighters while the ground troops fought droids in the city and the swamps.
Let us also not forget your Commander Locke example is proven wrong as it is the prophecy that saves the day not his personal actions.
Of course he was proven wrong; he was against the Protagonist and him being special. However, the movies were better for his existence. It created a clash of ideologies and personalities between Morpheus and Locke. It made sense that there were people, including some in authority, that didn't buy into the idea of the Prophecy and the One and wanted to focus on fighting conventionally, particularly because that is what he knew best.
TL;DR
The new about an intelligence guiding the Reapers is big news. The reaction to it is perfectly reasonable. The Reapers which are the tools of the intelligence are the main threat. Figure out a way to destroy or disable the tools and the intelligence is no longer a threat and can be dealt with at a later time. But the main priority now is finding a way to destroy, disable or at least combat the Reapers in some way shape or form that allows for some chance of victory to happen before any other move is made.
It's not big news. Nobody cares. The story doesn't care until the Catalyst just shows up. And then his only purpose is to put the ending choices in context and explain them.
The bold is a game too late and could have been solved by finding and dealing with the intelligence in some way for all they knew.