Using Bayesian methods to learn about our experimental effects is very interesting to me. For example http://www.evanmiller.org/bayesian-ab-testing.html, which may be the type of bandit you refer to.

In the best case, it might be more robust to low-data outliers that cause random blips that look significant, as well as addressing the problem of looking at an experiment over and over (which frequentists are uncomfortable with: more chances to succeed).

However, it is non-trivial to understand how this works for us in practice. Examples: we want overall results as well as days-in results (to look for novelty effects); we would have to choose priors with consideration, because they have a huge impact on most results; etc.

This takes time and effort, and there are a lot of other things competing for those resources.