No, it is required if we use the size of a Mass Relay and how far it can send a ship as a baseline. A Mass Relay has an Eezo sphere of five kilometers, and can send a ship a few thousand light years. For simplicity's sake, let's say 5,000 light years. The Andromeda galaxy is 2,538,000 light years away. So we would need something about 507.6 times the size of a Mass Relay. That comes out to an Eezo sphere alone being about 2538 kilometers, or 1577.04 miles across. That's an Eezo sphere about 56.78 times the size of the Citadel. And then you have to build the core to utilize that sphere and then a ship around that Eezo core. Not to mention the more mass you add, the more Eezo you need to move it. The resulting size of the ship is staggering, dwarfing anything any civilization, even the Reapers, ever made.
Nah, you misunderstand how mass effect physics works, and you're confusing mass relays with the drive cores of starships. They both use the mass effect, but for different things.
Mass relays create a corridor of completely mass-free space, which allows anything inside it to cross a vast distance instantly. This requires a vast amount of eezo and the mechanism has never been repeated except once, by Protheans with the Conduit.
Starship drive cores for FTL work by lowering the mass of the ship in a "bubble" to the point that it can travel faster than light. The amount of eezo only needs to be large enough to lower the mass of the ship enough, and for a long enough time. The ship wouldn't even need fuel once it attained maximum speed due to the force of inertia in space (and Newton's laws!). It only needs to maintain the mass effect field to keep the ship's mass lowered.