Whenever “it doesn’t matter too much what the prior is”, it makes more sense to use frequentist methods, which rely on large amounts of data to converge to the right solution.
… but only when the frequentist methods are easier to get working than the Bayesian approach. Even in large sample settings, it doesn’t make sense to give up the nice things that Bayesian methods provide (like coherence, directly interpretable credible intervals and regions, marginalization for nuisance parameters, etc.) unless the tradeoff gives you something of value in return, e.g., a reasonably accurate answer computed much faster.
… but only when the frequentist methods are easier to get working than the Bayesian approach. Even in large sample settings, it doesn’t make sense to give up the nice things that Bayesian methods provide (like coherence, directly interpretable credible intervals and regions, marginalization for nuisance parameters, etc.) unless the tradeoff gives you something of value in return, e.g., a reasonably accurate answer computed much faster.