Where in this picture does formulating the problem in a Turing-complete language help anyone? If making the "compiler"/analyser is (for practical/interesting purposes) impossible anyway?
I have spent several years in my PhD trying to map out just one probability distribution and make it computationally tractable. Postulating the existence of a "magic" analyser that work is redundant, which would be nice, but I don't see it ever happening.
Granted it has a billion input variables, but so does any "real-world", non-toy computer program.
TL;DR if the problem fits stabdard MCMC or rejection sampling just do it. If not then a magic performant analyser is going to be hard to build. I hope I am wrong :)