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 some 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.