Evolutionary models that make use of site-specific parameters have recently been criticized on the grounds that parameter estimates obtained under such models can be unreliable and lack theoretical guarantees of convergence. We present a simulation study providing empirical evidence that a simple version of the models in question does exhibit sensible convergence behavior and that additional taxa, despite not being independent of each other, lead to improved parameter estimates. Although it would be desirable to have theoretical guarantees of this, we argue that such guarantees would not be sufficient to justify the use of these models in practice. Instead, we emphasize the importance of taking the variance of parameter estimates into account rather than blindly trusting point estimates - this is standardly done by using the models to construct statistical hypothesis tests, which are then validated empirically via simulation studies.