They don't need to run for centuries. Hit maximum attainable FTL speed and "drift" whilst still accelerating due to Newton's law.
Also the pilgrims can use Prothean stasis tech to survive the centuries long journey.
As for breaking, that can be done in small increments months in advance of "touchdown". (clearly achievable with FTL drives)
So what's the problem...besides personal dislike?
Actually, Mass Effect drives dont work like that. Even in the fictional mass effect physics, what you are proposing violates conservation of energy.
The mass effect drive core maintains the region of decreased mass around the spacecraft. This allows the FTL travel. As soon as the drive core shuts down, the mass effect envelope collapses, and the ship returns to sublight velocity.
Remember, the ship isn't actually travelling FTL. It is only effectively travelling FTL. Even in a mass effect bubble, it is still travelling a tiny fraction of the new speed of light (which is raised within the mass effect field). When it collapses, it merely continues through space at the same fraction of c in a vacuum.
So unfortunately, your idea wont work.