The idea that the ark leaves after ME3 is even more stupid than the ME3 ending, and I need only one reason to know this, though many other good reasons have been provided.
ME:A is Bioware running away very quickly from the ME trilogy debacle, and the only way to do that is to completely bypass the endings. How do you do that? Leave before the endings.
Or leave after - a while after, mind you - and suddenly the time, and distance that the move to Andromeda provides, allows Bioware to handle the endings without them permeating every last aspect of the game.
Bioware likes continuity. They like respecting your choices - at least as far as their resources enable them to. If they didn't, we wouldn't have had the thousands of little touches that personalised ME3 to your trilogy playthrough. So I can't believe that they'd sweep under the carpet the one choice with the largest set of consequences.
Yes, they're putting some distance between ME:A and the ME3 endings. But that doesn't mean they're bypassing them entirely.
And I'd appreciate it if you didn't call me stupid for liking the idea that they're not, thank-you.
On top of that, if it's after ME3, there is absolutely no motivation to go to Andromeda. The only reason to escape to Andromeda is to flee impending doom, just in case. The only argument against this second point is if they aren't leaving for Andromeda, but they hit some wormhole and OOPS ANDROMEDA. Again, this is even more stupid than the ME3 ending.
I can provide an easy motivation - they found a shortcut. A relay, not built by the Reapers, travelling to Andromeda. If the Council found a relay travelling that far, you can bet that they'd want to know what's on the other side, who built it, and why. Especially with the lessons learned regarding the Citadel trap. And an exploration mission could quickly develop into a colonisation mission when they found free space on the far side.
Heck, an Andromeda relay is probably the second most popular fan-method of getting to Andromeda, behind the super-FTL Ark Ship. I'm just arguing that such a relay provides a motivation for going to Andromeda regardless of when it is found.