To me, it really depends on what you are trying to sample from, ie what exactly is the target posterior predictive distribution.
In particular, you have y \sim \pi(.) where \pi(.) could be factorized into a conditional via the product rule. Say you have a linear regression where y \sim \text{Normal}(X\beta, \sigma), you do need the joint posterior distribution of \beta, \sigma to generate accurate posterior predictive samples, but the hyperparameters of \beta, \sigma is not needed, which means you can take the marginal of them - in practice, it would mean using the MCMC samples of \beta, \sigma is sufficient.
So i would say no to both of your question above, unless the posterior samples are mostly independent as you noted. At the very least, you need the joint posterior from the lowest hierarchical.