For similar displaced volume, why does extra beam sharply improve a cruise ship’s roll stability?

Answer: Wide waterline leverage

Deeper hull leverage — Not quite. More draft can help some hulls, but depth is not the only way to create restoring leverage. For initial roll, naval architects care about how far the buoyancy line can move sideways as the ship heels. A broad waterplane gives that sideways motion a long lever without making the hull much deeper.

Wide waterline leverage ✓ — Right. The metacentric radius depends on the waterplane's moment of inertia divided by displaced volume, and beam is an outsized part of that geometry. Britannica notes that, for a given length and underwater volume, this term is proportional to the cube of beam. That is why adding width can buy a lot of roll stability while keeping draft shallow enough for ports.

Fin stabilizer lift — No. Fin stabilizers can reduce rolling while the ship is moving, so it is easy to over-credit them. But they do not provide the ship's static waterline geometry or replace intact stability. The beam effect exists before active fins do anything: a wider waterplane changes how buoyancy shifts during heel.