The more I think about it, the more I think the screw up started with ME2.
The collectors should not have been building a human reaper, they should have been building a 2nd Citadel, so the Reaper can come back from dark space. Shepard and team only partially succeeds in stopping them at the end. The 2nd Citadel is destroyed but the the Mass Relay part of it works long enough for all the Reapers to come through.
Shepard and company beats a hasty retreat to the Omega-4 Relay, and blows the relay up, in order to prevent the Reapers from coming through and quickly reach the rest of the Galaxy. Now the Reapers have to physically travel to the next Mass Relay, which will take a year or so, this buys the time for Shepard to try to warn the rest of the Galaxy at then end of ME2.
Also, while destroying the half finished Citadel, Shepard finds the construction plans for the 2nd Citadel, and it is through the reverse engineering of that plan that humans reach a greater understanding of what the Relays are and later come up with the design for the Crucible super device to fight the Reapers. The Reapers attack Humans first because this knowledge makes humans the most dangerous so they want to wipe human's out first. Shepard spends ME3 trying to rally the Galaxy and get all the species to help build the Crucible. Cerberus believes that humans can build it alone and does not want to share the knowledge with the rest of the galaxy, so that after the Reapers are beaten humans and humans alone can dominate the galaxy rather than share power with the rest. Players can go Paragon (work with galaxy), or Renegade (work with Cerberus).