It is possible to do hypothesis testing, regression, classification, ect, all using Bayesian methods. Furthermore, these Bayesian methods are more flexible and easily interpret-able than Frequentist methods. Therefore, why solve a statistical problem in a more difficult manner? Really, the true reason why frequentist methods were standard is issues of computing. With lack of computer the Bayesian approach was not feasible. Now it is no longer an issue. Therefore, why would anyone still rely on frequenstist methods? Furthermore, still universities still teach frequenstist methods, or entirely abandon them? Look at it this way, the slide-ruler was once taught in universities, now nobody teaches it anymore since there is a better alternative. So why continue to use, and to teach frequentist statistics?
The only reason that I can think of is that frequentist methods are computationally faster (substantially) than Bayesian methods. For example, it is possible to do simple linear regression with least squares with thousands of predictor variables and a dataset in the trillions, very quickly, but that task might be too long to calculate for MCMC.
Edit: It says "question closed" because "it is not focused". This objection to close the question is non-sense, there is only one question being asked, namely, "why would anyone use a frequentist approach when it can be solved in a Bayesian manner?".