Hi @Jack_Caster,
The clearest discussion of this that I’ve found is chapter 13 of Gelman and Hill.
Since first posting this, I’ve gained confidence that I’m understanding it correctly, i.e. that
by not modeling the covariance, we’re in effect using an infinitely strong prior on the coefficients being independent
… and that if you’re using a model for prediction, you may be losing predictive power by not modeling the covariance (e.g. the attacking/defending strengths correlation example in my post above).
It may be ok for your problem if it you have no prior reason to believe that the group-level coefficients are correlated.