This question was raised before here Covariance of multiple varying coefficients in hierarchical models and I meant to write down some though but never manage to. It is something used to confuse me as well, and now my understanding is quite similar to yours.
Classically, the covariance of the betas/coefficients (regardless it is for the fixed effect or the random effect) usually comes into play when you are computing the contrast or linear combinations among them (eg. ANOVA). This process itself is projecting the contrast to some space using the projection matrix (rotation and scaling etc) derived from the covariance matrix. In Bayesian LME, since the posterior samples already in the right space, we can compute the contrast directly using the sample to have the correct marginal. But if we model the covariance specifically, then we should rotate the posterior when we are computing different contrast.