That's not true at all, nor is it applicable to this hypothetical situation. Taking the beaches was required to take Berlin. The point of going after the Intelligence would be the hope that taking it out would stop the Reapers and you wouldn't have to fight all of them, like when the Control ships are taken out in The Avengers or The Phantom Menace.
The intelligence is sitting undefended on the Citadel, which is not captured by the Reapers until the end of the game. I don't see anyone rushing to stop Shepard from shooting the tube. They don't know this at the time, but it's no reason to not look into it. Leviathan shows they have teams on special projects other than the Crucible.
Who said they should drop everything? Can you just once argue against what I said rather than a strawman?
The Crucible is unknown and unproven. While anything should be pursued at this point, why put everything into it and nothing else? Where's the Commander Locke character from The Matrix Reloaded and Revolutions who doesn't believe in the prophecy and is doing everything he can to win in a way he knows how? That's some compelling conflict.
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.
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.
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.
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.