Oh it's *enormously* contrived, but that's besides the point in this instance. The developers did play fair by establishing up front that there's no way even a united galaxy could take on the entire reaper fleet and win. Taking down one or two by throwing whole fleets at them sure, but they have more Reapers than you'll ever have fleets. It's a war of attrition that simply cannot be won.
Yet the only way to beat them is with the God gun that it knows about....much is made of this cycle being different, and Humanity in particular being the glue that holds it all together, yet effectively we just pick up the ball that the previous cycle had dropped and get it over the line.