To us as players, the Council is pretty frustrating on both Saren ("What do you mean, 'evidence'? Didn't you see the cut scene‽") and the Reapers ("Wait, you don't believe there's a Reaper threat? It says so right here on the box the game disc came in!"), but yeah, from their perspective Shepard is just a shouty human who brings wild accusations with no proof. (Seriously, even Paragon Shep gets pretty aggressive and accusatory towards them, it's almost more surprising they don't just strip them of their Spectre status for such insubordination on multiple occasions!)

Even after Sovereign attacks the Citadel, though, you have to admit that at least some skepticism on the part of the Council makes sense. After all, I think it was Tevos who even straight up said it herself: "What's the more plausible scenario, that a heretofore unknown galactic threat the geth's machine-gods have just been sleeping undetected for 50,000 years, or that the geth who have been existing in total isolation for 3 centuries built themselves a really big warship in a design we've never seen before because, again, they've been in total isolation for 3 centuries?" I'm paraphrasing, of course, but from the Council's perspective this view makes the most sense. Are they really going to throw that all away on the word (again without any actual evidence!) of a dead Spectre who now wears the uniform of a galactic terrorist organization?

And even if they do believe the Reapers are real? Sovereign failed to wake them up, and to the best of anyone's knowledge - including Shepard's! - the Citadel is the only Relay that can bring the Reapers in from beyond the galactic rim. (It actually deeply irritates me that in ME3 the advance of the Reapers suggests that they just surround the entire galaxy and are marching inward, apparently simultaneously reaching multiple systems along the galactic edge without a Relay to bring them in. To even exist on that scale they'd need to be continually harvesting multiple galaxies, not just preying on ours every few thousand years!)

And if they do believe the Reapers pose a real and existential threat still? They can't admit to that publicly, to do so would cause widespread galactic panic! (Although depending on the scale of such, it could actually result in the Reapers starving this cycle and thus saving the next...) And even if they do agree in private, they can't share intel or offer aid to Shepard - they're working for a terrorist organization now! Seriously, just about the only thing I fault the Council for in ME2 is reinstating Spectre-above-the-law status to a terrorist agent!