What complex situation you are referring to? I've used it in robot localization/filtering algorithm, and things can be simplified by assuming the process is stochastic.
I think the problem is that the real application of Bayes theorem is to convert P(A|B) to P(B|A), and the visualisation doesn't map well that. You have
P(B|A)P(A)
P(A|B) = -------------
P(B)
But there is nothing in the visualisation that helps you see what those ratios are ... even less so what it might mean to multiply P(B|A) and P(A).
Venn diagram shows the set of outcome of an experiment. But in many real world situation the set of outcome of an experiment is too large or difficult to work with. I think Venn diagram approach is very helpful if you finite set of outcome of an experiment that you can write/generate and draw/color those sets.
It's right that P(B|A)P(A) is not intuitive, but if it is replaced by P(B cap A) or P(A cap B) suddenly you are back to the definition of probability of an event. Calculating P(B) using the law of total probability was very illuminating for me.
