The Normandy had them installed because Andeson had planned to use it as a mobile command center for the war effort. The fact that there were only a few on the whole of Earth doesn't actually argue for their non-rarity.
It's just an idea. They would need to capture one to track the other end, presumably. But why would you think the people that installed such things on the Normandy are the same ones that woluld be in charge of the Ark? Then again, the Reapers might have just destroyed the Milky Way side of the connection.
Really? You are honestly telling me that you think they wouldn't install this established, available and - especially for these particular circumstances - incredibly useful technology on such a ship, when they do it pretty much everywhere else?
As I said, I think they only way to to make it feasible is to break the dam thing, however they are going to do it, it's just going to be yet another contrivance.
My point is also not about the QECs specifically, that's just a symptom of the larger problem, I see with the galaxy change. And that is - as Iakus said already - that the scenario comes out of the blue, without any kind of setup. Yes, a clean slate is nice but the point of a clean slate is to finally enable consistency in the universe again and reduce the countless contrivances that have plagued this franchise ever since ME2. It does not bode well if this is already dampened by the very premise of that clean slate. An ark project like this does sound very familiar and immediately reminds me of things like the Lazarus Project, the Cerberus superpowers, the crucible and the catalyst, i.e. an obvious and cheap plot device that comes out of nowhere.
Not a good start IMO, if the rumors are true.