The reason A/B testing has many parameters is because at the end of data collection it hopes to employ a hypothesis test to determine whether variation was better performing as compared to control (the interpretation of "better" is done according to parameters defined). The confidence level and other parameters allow you to do proper risk-reward analysis and see if you want to go ahead with variation. Moreover, minimal sample size (visitors/conversions) ensures that local (initial) fluctuations do not unnecessarily bias the test results. In fact, in A/B test first 100 visitors are virtually identical to last 100 visitors with regards to impact on final results.
However, I am guessing in bandit algorithms (since their primary usage is continuous optimization and not hypothesis testing) local fluctuations can have an adverse impact on ultimate results which may be okay for continuous optimization problems but not for projects where you need to determine with a defined level of confidence whether one version works better than the other.
Different needs, different algorithms.