This paper is devoted to the convergence analysis of the generalized Robin-Neumann schemes introduced in [Internat. J. Numer. Methods Engrg., 101(3):199–229, 2015] for the coupling of a viscous incompressible fluid with a thick-walled elastic or viscoelastic structure. To this purpose, a representative linearized setting is considered. The methods are formulated within a class of operator splitting schemes which treat implicitly the coupling between the fluid and the solid inertia contributions. This guarantees energy stability. A priori error estimates are derived for all the explicit and semi-implicit variants. The analysis predicts a non-uniformity in space of the splitting error, hence confirming the numerical evidence of [Internat. J. Numer. Methods Engrg., 101(3):199–229, 2015] for the explicit variants. Besides, the analysis demonstrates that the genesis of this accuracy loss is the spatial non-uniformity of the discrete elastic or viscoelastic solid operator. The theoretical findings are illustrated via a numerical study which shows, in particular, that alternative splitting schemes recently reported in the literature also suffer from these accuracy issues.