As both a Mass Effect and Stargate fan, I approve of this thread! You and the OP both have good ideas here, but I don't think the 600 year thing works if you plan on involving the Milky Way again. Too much can change. Maybe they create a temporary wormhole, and the only way is harnessing the power of a star in such a way that the star is destroyed in the process (very Stargate-y).
I like the idea of each ark carrying a piece of the relay that you have to reassemble later.
Game 1 could be about exploration -- learning about the Andromeda galaxy, the cultures, the Kett enemy, and maybe finding the second piece of the relay.
Game 2 could be finding the remaining pieces of the relay and establishing outposts. End it on a huge battle of Ryder defending the relay construction site as the engineers complete its reconstruction. The battle doesn't go well and the Kett take possession of the relay. Ryder is forced to retreat on the Tempest, but not before sending a probe with a message (and let the player compose the message, sort of like the suicide mission final speech) through the relay to warn the Milky Way. Depending on what you say in the message it could affect the Milky Way response in game 3.
Game 3 has some returning characters from the OT (depends what they make canon). Perhaps part of it takes place in the Milky Way, even if it is just some cinematic of people viewing your warning message. This lets you visit some familiar landmarks (maybe even the citadel) but the main focus is still Andromeda. Eventually you lead reinforcements (though not as much as you'd like, since they're still recovering from the Reaper war) to take back control of the Andromeda relay and end the Kett threat. Imagine if you actually got an OT squad mate or something.
Biggest problem with involving the Milky Way is some players won't be happy with their personal ME3 ending not being canon. Ultimately it doesn't matter since this is all a pipe dream. But yeah, Andromeda really should have been the Atlantis of the franchise. Perfect comparison.