Hibernation/cryosleep + dischargeless drive core (or detachable battery storage for discharge) = perfectly valid way to get to Andromeda, given there's no time restrictions. The only thing you have to deal with is how and when the Ark itself is constructed, and that's massaging the timeline, not the lore.
No, you also have to deal with energy. Mass Effect technology doesn't run on the kind of convenient infinite energy power cells your little scenario requires. Eezo power drain for 570 years > power storage for 570 years > ship has to be bigger to accomodate for all that power storage > eezo power drain increases > energy storage has to be bigger > ship has to be even bigger, and so forth. The design of a FTL generation ship is inherently doomed to fail because unlike relativistic generation ships, which drifts through space with minimal power requirements, FTL generation ships need to keep their FTL engine on for the entire journey. That's like building a car engine that has to run at maximum speed for 570 years without being refuelled. The engine would have to be unbelievably huge to accomodate for the fuel storage - the same principle applies here.
This isn't even touching the issue of maintenance or discharging, the latter of which you completely dismiss as doable with modern technology despite the fact that such technology doesn't exist for spaceships. Only extremely large, extremely power-hungry and extremely maintenance-heavy space installations have the capacity for their own discharge technology, and these installations are completely reliant on fuel and power supplies from the outside to remain operational. I shouldn't have to mention this, but these kinds of installations obviously don't have FTL propulsion since the enormous mass would require a proportionally enormous amount of eezo and power to reduce, on the scale of economical and practical impossibility. It's the equivalent of building a floating city that's as fast as a racing boat.
A generation ship would be completely cut off from the galaxy it left. There would be no way to bring the necessary supplies it needs to continue the journey. Everything it needs to make the journey, it needs to bring with it right from the start. It can't even use solar panels to generate its own power because the panels doesn't get any useful amount of light in dark space. And other methods of self-sustained power are either insufficient, impractical or extremely hazardous.
This is why the Ark theory doesn't work.