Fanning out across the galaxy got more Reapers killed than they had in most cycles, even without the Crucible.
Most cycles they don't lose even a single Sovereign-class
This cycle is also different in that the Reapers thought they knew they were coming. Everyone would have time to prepare; fanning out at once before they have any more time to prepare, surprising the daylights out of them, makes more sense than systematically working through each system and giving all others even more time. Mix that with any sort of suicidal hopelessness, and you've got the potential for even more dead Reapers.
They control the relays. They can open and close whatever ones they wish. Like they've done in every cycle before this.
This cycle has already demolished one relay for the sake of inhibiting their travel. How do you think folks are going to react when they discover the Reapers are gaining control of the Citadel and then, pop, the relays go dark?
And the purpose was...what? to get a couple of apes onto the Citadel and frak up the Catalyst's plans? 
Transporting human bodies, remember?
And yet they didn't. Did the Catalyst only have three husks and a marauder available to repel borders?
At that exact point, it's possible. Whether Shepard lumbers forward takes out the three musketeers and Marauder Shields or quadruple that number, the game continues forward.
Plus, they had Harbinger front and center until the threat was (supposedly) neutralized. They underestimated CyborgShep.
Which in two thousand years had never been activated until that very moment? 
Nope, not under these circumstances.
Except it shouldn't have been a contest. The Reapers should have won by slaughter rule. Literally. The deck was stacked so badly the only way the Reapers could have lost is if the Catalyst was suicidal. That it wanted Shepard to kill it from the very start.
Had it not been for the Crucible, every single thing they did would have still succeeded despite the Protheans' warning and Keeper trick, which indeed changed the variables.
Yup. Why would the organics believe any of this? Why should Shepard believe any of this?
Significantly easier for Shepard to believe what's in front of him than for the organics to believe: "Hey, guys, STOP FOR A MINUTE. THE REAPER OVERLORD SAYS IT'S OKAY."
Shepard: "Hackett? Patch me in to the Crucible specialists. This glowing child of a little boy I saw die a few months ago claims that shooting this pipe will activate the Crucible and end all synthetic life in the galaxy. I need to know if it's just blowing smoke"
Which would lead to a technobabble version of: "It's possible." Yay.
Yeah, phoning a friend or developing a consensus, or at least doing some due diligence might be a little bit important when you're reshaping the galaxy.
Why? What good would that do? You think they'll actually reach an agreement between the solutions?
You know everything you need to know beforehand, and you don't need the indecisiveness of a desperate, thinning galaxy to help you decide.
Especially since your only source of information is a being with every reason to lie, has mind-control powers, and has been committing extinction level events for a billion years or so. Oh, and is operating on faulty logic to begin with.
Like you said, it's got this thing in the bag. Why would it lie? I understand not trusting its agenda, but it has no reason to lie about the device's capabilities.
Mind control? Perhaps, but it's somewhat
common knowledge that it takes time for Reaper indoctrination to set in, and mind-control could be going on from any number of the Reapers outside the Crucible at that point.
Extinction-level events? True, to exert order over the chaos of organic evolution, only when civilizations are at the apex of their glory.
Faulty logic is entirely subjective.