There is a whole galaxy full of resources to access and millions of other soldiers and Marines serving in the Council species militaries. There were would be no need to give Shepard and the Normandy the responsibility of tracking down resources for the Arks when there would be other people and other ships much better suiited for that role. Shepard already had a mission on his or her plate that took precedence over the ark project. Relying on Shepard would both be an inefficient use of manpower and a security risk.
Shepard also isn't personally gathering resources. At best a probe from the Normandy flags an asteroid as being eezo rich, and then the Alliance or Council sends a mining operation to tow or extract it. Shepard doesn't need to know whether that resource later goes to the Crucible, war ship production, the arks, ect.
Then why the hell is Shepard referring everyone to Docking Bay D24?
What have I been stashing there and handing out to everyone on the Citadel, the Normandy's dirty laundry? 
There are a non zero number of people who believe we've been on Mars for years. Some people even claim to have been teleported there as part of a secret government project. 
More seriously, when was it strongly hinted that the Reapers had nothing to do with this?
Mac Walters has said that people are boarding the Ark for a variety of reasons. If it was because of the Reapers, I would think the predominant reason would be "AHHHHHHH!!! REAPERS!!!! WE CAN"T DEFEAT THEM CONVENTIONALLY!!!!!!"
Uh, do you realize how many people would want a seat on that ship if it were made public?
Even with Elon Musk predicting that the crew of that first voyage will probably die?
At any rate, I imagine the ship would be crewed by people who serve important functions on the ship or the colonization efforts, and wouldn't be a passenger liner. Crew would be drawn from people with those skills. And they will have trained and prepared for years for the program.