In ME1 the Council acted in a completely reasonable fashion regarding Shepard and his Reaper and Saren warnings. However, Shepard Can't Be Held Down By The Man, so BioWare tries to present this as a bunch of obstinate old politicos who can't see the truth as revealed in cutscenes where neither they nor Shepard was present.
With regards to Saren, a young Alliance soldier comes in claiming that the top Spectre operative just organized a terrorist attack against a peaceful colony with an army of robots that haven't been seen in civilized space in hundreds of years. His evidence: a dock worker and criminal, who was asleep during the attack, said during a gunpoint interrogation (being held by the same soldier bringing the charges) that a turian named Saren shot Nihlus. WELL THEN OPEN AND SHUT CASE. Also, the soldier's mentor is a man who has a long-standing grudge against Saren and apparently a history of accusations that have never been proven.
Now, you, the glorious Shepard-Conduit, the Player, saw that cutscene where Saren shot Nihlus and rode away on the monorail and then shouted at Benezia in that weird room in Sovereign (where was that, why was there a room with a table and chair in there). Why can't these fools see the truth (that nobody saw)!?
Also it doesn't help that during the meeting, Shepard's option is to A) claim it's Saren's racist conspiracy against poor humans, B) make a smug jackoff statement about how the Council's already decided, or C) basically tell the jury in your court case that they're idiots. Nice military professionalism there, douche.
But alright then, you guys want proof then we'll give you proof an- oh they accept it unconditionally about 2 seconds after they hear it, which actually seems kind of crazy but whatever. And then they make you a space commando in a great honor for your species after about a minute of thinking it over.
Well... they're still jerks, because they didn't believe your haphazard conclusion that a magical bad dream you had and a word in the recording mean that the Protheans were wiped out by ancient machine gods that are coming back right now. This theory, which takes Shepard about half an instant to come up with and accept whole, is of course correct and the Council is stupid for not accepting your apocalyptic ravings and hallucinations as ironclad evidence that they should prepare their quadrillions of citizens for galactic war.
And in the end, they finally accept that the Reapers are real, and basically kiss your feet and vow to work together with you to stop them.
Now I'm exhausted, but basically in ME2 it's dumb because it takes that last part and just goes "lol jk the council doesn't believe you" even when there's absolutely no real reason for them to not believe you except to once again establish Shepard as a misunderstood prophet even when the entire galaxy saw him being proven correct two years before.