Spatial discretization techniques for physical systems often give rise to large linear systems of equations whose solution dominates the overall computational cost. Consequently, large-scale simulations rely on sophisticated preconditioned iterative solvers. Their performance is governed by a multitude of interacting algorithmic and numerical parameters. In practice, these parameters are frequently selected based on heuristics or limited parameter studies, despite their strong impact on time-to-solution — particularly for coupled multi-physics problems. In this contribution, we demonstrate how global sensitivity analysis can be leveraged as a tool to systematically identify performance-critical solver parameters in computational mechanics applications. Focusing on block-structured linear systems arising from finite element discretization of coupled multi-physics systems, we treat time-to-solution itself as the quantity of interest and measure how parametric choices in algebraic multigrid and block preconditioning (such as smoother properties and parameters controlling the approximate block solves) contribute to variability in computational performance. The methodology combines the learning of a data-driven surrogate performance model from a structured sample of solver configurations with the subsequent computation of Sobol indices for the solver parameters. Rather than emphasizing sensitivity analysis as a methodological end in itself, the talk highlights its role as an analytical tool for complex numerical workflows. Through representative multi-physics benchmarks, including fluid-structure interaction and contact mechanics, we show how sensitivity measures reveal dominant parameters and identify configurations that can be fixed to robust defaults without degrading performance. As an outlook, we argue that applying global sensitivity analysis to abstract models of numerical methods where inputs are algorithmic or numerical (rather than physical) parameters forms a broadly applicable paradigm for the computational mechanics community.