The critique of frequentist statistics, as I understand it - and I don't think I do - is that frequentists like to count things, and trust that having large sample sizes will take care of biases for them. Therefore, a case in which frequentist statistics co-occurs with bad results counts against use of frequentist statistics, and you don't have to worry about why the results were bad.

The whole Bayesian vs. frequentist argument seems a little silly to me. It's like arguing that screws are better than nails. It's true that, for any particular individual joint you wish to connect, a screw will probably connect it more securely and reversibly than a nail. That doesn't mean there's no use for nails.