Sorry for the delayed response, but I didn’t spot this thread had progressed.
I completely agree that there are significant challenges in developing and validating the performance of algorithms for models of this kind. Our current work aims to provide an environment where we can demonstrate the challenges (eg by having generated quantities for discrete variable models that allow us to see that Rhat is not what we want) and to develop novel numerical Bayesian inference algorithms that address the deficiencies of the current state-of-the-art in MCMC.
For what it’s worth, my fear is that there’s a lot of research going on at present looking to develop numerical Bayesian inference algorithms, but I think the world needs to get better at exposing the challenges users would like to see solved and therefore helping research to focus on tackling the “real issues” in a way that could migrate into future variants of Stan. That’s probably more aligned with another topic (here: Reimplementing the inference algorithms - Algorithms - The Stan Forums (mc-stan.org)).