I've rarely used FT methods in the real world (one reason being the wrap-around boundary conditions), and not really followed them. And, not sure whether that paper reflects the current state of the art.
The method in the paper exactly represents the solution as a Fourier-type integral which must be computed numerically. The error in the quadrature rule is the sole source of pricing error.
I guess whether the extra computational cost in computing those integrals exactly with the method the author used would carry a corresponding financial cost.