Hi @DanWeitzenfeld, I had a question similar to yours that we discussed in this thread.
As you did, I noticed that some hierarchical models make use of the covariance matrix (especially the ones implemented in STAN), while some others do not. I was wondering whether you found a clear explanation on this modelling technique. From my experiments, it seems that it does not affect much the fitting. To be fair though, the dataset I used has a correlation between group level predictors of about 0.06 (i.e. very low).