Frequentist statistics are just a special case of Bayesian statistics with certain implicit, built-in priors - if you pick these priors using Bayesian statistics you'll get the same answers.
The frequentest toolbox is just a collection of these useful special cases of Bayesian statistics with priors that usually make sense in practice. The advantage is this greatly simplifies the statistical analysis for many problems. Sometimes these methods fail though, when the priors they rely on implicitly are far off from the actual, and so a Bayesian analysis is needed.
Of course, sometimes it's difficult or impossible to use Bayesian methods.
It's analogous to classical vs. quantum/relativistic physics. For many cases, you can get the right answer to a problem using classical physics. But under certain conditions, classical physics breaks down, and you must apply quantum or relativistic physics to get a meaningful answer. On the other hand, for many problems it would be silly or impossible to use quantum or relativistic physics because classical is perfectly good.
So you might get into argument about whether a specific case can be adequately handled by frequentest statistical methods, or whether a bayesian analysis can/must be applied.
The philosophical debate about the different approaches to the nature of probability is just that - a philosophical one, and one that has no real bearing on the usefulness and correctness of bayesian or frequentist statistics in practice.