Correction, the ending shows the Crucible wave covering half the galaxy. That's how it is
OK, cool, it is possible, I accept that. But let's take a look from another side. Designing ark ships to preserve all Council races will take time, especially when built from a scratch. To have it done in two years will require 1) belief in upcoming Reaper arrival (which should be impossible since Shepard stopped Sovereign) 2) a lot of money and resources. Who has that much power in the galaxy? Let's assume volus. Why and how will the construction be organized in such a way that no one knows about it, especially when galaxy uses all the available resources and scientists to build the Crucible? Not even the Shadow Broker knows about it. Next, why would the builders bother with preserving all races? And how are they going to secretly collect people from Reaper-controlled systems?
Well, my personal version of Ark Theory actually involves a clandestine organization (Possibly with sympathizes in Citadel governments) either taking Shepard's initial warning to heart or knowing about the Reapers years before ME1. So they would have time to gather serious resources, possibly bartering places on the Ark for the expertise and materials they'd need. If the Alliance could hide the construction of the Crucible I think hiding the construction of a relatively speaking much smaller project isn't unreasonable. The only reason so many people knew Alliance was working on something was because of how ginormous it was. The Galaxy is a big place and it isn't impossible to keep secrets from the Shadow Broker (Case in point, the Thessia Beacon). Not every scientist and resource in the galaxy was working on the Crucible either. The Reaper War would be a kick in the pants to their development, not the start of it.
As to the passengers, they would have been assembling them for some time though there would be a rush to gather them during the Reaper's attack on the Galaxy certainly. Why preserve all races? Well, maybe the minds behind the project are just that idealistic.
Though there is another idea that the Ark settlers had planned to use the Citadel Relay to jumpstart their trip across the void and assembled their ships at the Citadel for that purpose. When the Reapers retook the Citadel just before Priority Earth, the Arks opened their doors to the people of the Citadel before making the jump (Which also opens the potential for returning characters that were on the Citadel)