1. When your main objective is not prediction but understanding the effect of some underlying / unobserved random variable.
2. When you don't have tons data + you have very clear ideas of the data generation process.
(1) is mainly relevant for science rather than private companies, e.g. if you're an epidemiologist, you're generally speaking interested in determining the effect of certain underlying factors, e.g. effect of mobility patterns, rather than just predicting the number of infected people tomorrow since the hidden variables are often someting you can directly control, e.g. impose travel restrictions.
(2) can occur either in academic settings or in private sector in applications such as revenue optimization. In these scenarios, it's also very useful to have a notion of the "risk" you're taking by optimizing according to this model. Such a notion of risk is in the Bayesian framework completely straight-forward, while less so in the frequentist scenarios.
I've been involved in the above scenarios and have seen clear advantages of using Bayesian inference, both in academia and private sector.
With that being said, I don't think ever Bayesian inference, and thus even less so PPLs, are going to "take off" in a similar fashion to many other machine learning techniques. The reason for this are fairly simple:
1. It's difficult. Applying these techniques efficiently and correctly is way more difficult than standard frequentist methods (even interpeting the results is often non-trivial).
2. The applicability of Bayesian inference (and thus PPLs) is just so much more limited due to the computational complexity + reduction in utility of the methods as data increases (which, for private companies, is more and more the case).
PPLs mainly try to address (1), and we do have examples of very successful examples of this, e.g. PyMC3 (they also have a bunch of nice examples of applying Bayesian inference in private sector context), and Stan (maybe more heavily used in academia).