The Council didn't believe in the Reapers when one knocked on their door and tried to kill them, with testimony from one of their Spectre agents backing up the idea. I'm not saying these people have anymore proof Shepard, in fact I'm imagining them as a bit of a fringe group, or a secret society, that used a substantial amount of personal wealth and whatever investors they could find, combined with a massive influx of investors looking for a way out of the Reaper War once it began. If something as massive as the Crucible can be completed in a few months, I imagine they can build several much smaller ark ships (Or retrofit existing ships) with years on their hands even with fewer resources.
I imagine it happened just before the Reapers retook the station. All the witnesses are dead (Or possibly some escaped on the Ark ships).
Its an interesting theory, but there is a big issue with the undertaking of an intergalactic voyage.
Mainly, how many people, and species are we talking here? Being able to have a stable population you would need at least several thousand members of a species to avoid too much interbreeding in subsequent generations. If this ark is to be the galaxy's attempt to survive the Reapers then you would want members of every species; unless we are dealing with another _____ (insert species name) supremacist group that is only interested in saving their own people.
If this ark is supposed to allow all the races of the galaxy to survive the invasion then you would need at least several thousand members of the Human, Asari (less so since they can breed with anyone), Turians, Salarians, Quarians, Krogan, Batarians, Volus, Elcor, Vorcha, Hanar, Yahg and Drell species; and those are just the Council/Non-Council races. That doesn't even take into account other races like the Rachni or Geth, who could (potentially) be allies in ME 3 and would like to avoid extinction as well; though the Rachni would only need one queen and the Geth are software.
That's 15 species (each with their own viably sustainable populations) that the ark would need to transport, and that doesn't even begin to cover the other items that the ship would need to bring. Things such as supplies/colonization materials, power sources, etc. would also need to be brought along as well, meaning that this ship would probably be larger then a Reaper capital ship in size. Sure, the ark could save space by only carrying the genetic information of the races, and then cloning people once it got to it's destination, but really at that point you might as well just reboot the franchise.
Another thing to consider, if the species of the milky way do make it to the next galaxy over, is the viability of potential colonies. The Quarians and Turians have completely different food requirements then the other races, what do they do if there are no compatible worlds?