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We study the properties of the strongly-coupled quark-gluon plasma with a multistage model of heavy ion collisions that combines the T$_\mathrm{R}$ENTo initial condition ansatz, free-streaming, viscous relativistic hydrodynamics, and a relativistic hadronic transport. A model-to-data comparison with Bayesian inference is performed, revisiting assumptions made in previous studies. The role of parameter priors is studied in light of their importance towards the interpretation of results. We emphasize the use of closure tests to perform extensive validation of the analysis workflow before comparison with observations. Our study combines measurements from the Large Hadron Collider and the Relativistic Heavy Ion Collider, achieving a good simultaneous description of a wide range of hadronic observables from both colliders. The selected experimental data provide reasonable constraints on the shear and the bulk viscosities of the quark-gluon plasma at $T\sim$ 150-250 MeV, but their constraining power degrades at higher temperatures $T \gtrsim 250$ MeV. Furthermore, these viscosity constraints are found to depend significantly on how viscous corrections are handled in the transition from hydrodynamics to the hadronic transport. Several other model parameters, including the free-streaming time, show similar model sensitivity while the initial condition parameters associated with the T$_\mathrm{R}$ENTo ansatz are quite robust against variations of the particlization prescription. We also report on the sensitivity of individual observables to the various model parameters. Finally, Bayesian model selection is used to quantitatively compare the agreement with measurements for different sets of model assumptions, including different particlization models and different choices for which parameters are allowed to vary between RHIC and LHC energies.