Given that it floats, we know that its own volume is larger than the water it displaces.

And this is where you went wrong; if the volume of the boat is larger than the water it displaces the boat will not float (assuming the boat's material is denser than water).

I don't follow. Making up numbers here. 1 cubic meter of Styrofoam weighs 1 kilogram. Therefore it displaces 1 kilogram of water. 1 kilogram of water is 1 liter, which is less than 1 cubic meter, so the Styrofoam floats. The volume of the Styrofoam is larger than the water it displaces so it DOES float. That's the opposite of what you said.

Not trying to argue, just very confused...