The complete numerical analysis of time splitting schemes which avoid strong coupling has rarely been addressed in the literature of unfitted mesh methods for incompressible fluid-structure interaction. In this paper, an error analysis of the semi-implicit scheme recently reported in [Int. J. Numer. Methods Eng., 2021; 122:5384--5408] is performed for a linear fluid-structure interaction system. The analysis shows that, under a hyperbolic-CFL condition, the leading term in the energy error scales as $O(h^{r-1/2})$, where $r=1, 2$ stands for the extrapolation order of the solid velocity in the viscous fluid substep. The theoretical findings are illustrated via a numerical experiments which show, in particular, that the considered method avoids the spatial non-uniformity issues of standard loosely coupled schemes and that it delivers practically the same accuracy as the strongly coupled scheme.