Multilevel models are one example of problem were Bayesian methods are hard to avoid as otherwise inference is unstable, particularly when available observations are not abundant. Multilevel models should be used more often as shrinking of effect sizes is important to make robust estimates.
Lots of flashy results published in Nature Medicine and similar journals turn out to be statistical noise when you look at them from a rigorous perspective with adequate shrinking. I often review for these journals, and it's a constant struggle to try to inject some rigor.
From a more general perspective, many frequentist methods fall prey to Lindley's Paradox. In simple terms, their inference is poorly calibrated for large sample sizes. They often mistake a negligible deviation from the null for a "statistically significant" discovery, even when the evidence actually supports the null. This is quite typical in clinical trials. (Spiegelhalter et al, 2003) is a great read to learn more even if you are not interested in medical statistics [1].
[1] https://onlinelibrary.wiley.com/doi/book/10.1002/0470092602