I really doubt the asari knew about the reapers.
Here's a theory I posted about the asari and Andromeda
For now, I will say the ship leaves before the events of ME3
This solves a lot of problems. The only problem I could find is that they needed a drive that could get to Andromeda without discharging. Now if they'd discovered a derelict proto-reaper it would be possible. This ship would not contain the reaper core that does the indoctrination. It would be one of the ones from before Harbinger, the first true reaper, was built. With its onboard AI completely destroyed in the battle with the Leviathans, all that is left is the drive core, the shell. Perhaps the weapons were destroyed, too.
The Asari have absolutely no clue as to where it came from, only that it's well over a billion years old and that whatever built it is long dead. They can't reverse engineer it, but they can recondition it and build a new ship around it. The true reaper hull came later. We can say that in new lore. This drive does not need to discharge.
The reason I've been advocating the Asari based project is because they are probably the only species who would actively invite the other species to participate due to Asari reproductive preferences - since they discovered space flight they have preferred to mate with other species. This has also put them in the position of being the lead species on The Council, trying to maintain peace. War between the galactic factions threatens their supply of genetic imprints. Peace is for strictly selfish reasons. There is a delicate balance in the galaxy, and that's why that archive was kept secret. The Asari had to have the galaxy believe they accomplished everything on their own.
So all of this takes place centuries before the events of ME1. Humans get there before the events of ME1. We even have Quarians there without their suits. And no Geth needed.
This solves the problem of the all the events and choices throughout the Mass Effect series up to and including the ending.