The answers are different in that they highlight different aspects of the subject. However, their implications do not conflict. Sequential modelling of the conditional mean and variance can be justified when the conditional mean part can be estimated consistently even in presence of GARCH errors. AR-GARCH model would be one such example. Meanwhile, it is more difficult to justify sequential modelling when neglected GARCH errors ruin the consistency of the conditional mean estimation. ARMA-GARCH with non-empty MA part is one such example.
In practice, you may of course try selecting the conditional mean model ignoring the GARCH structure in the errors and see what happens. This is an easier setup and requires less computing. However, you have no guarantee over how much the selected model will differ from a model selected incorporating the information on the GARCH structure. It is only fair to admit that even considering the conditional mean and variance models simultaneously does not guarantee "perfect" results. It is just that this approach is easier to justify based on properties of the resulting estimators.
Putting diplomacy aside, why don't you try it your way and see if the results are satisfactory. You could hide some data from yourself in a hold-out sample when selecting the model. When the "optimal" model has been selected, you would test the model performance on it.