The other problem with the "Bayes with flat prior = frequentist maximum likelihood" idea is that, even if you ignore the issues with improper priors, the concept of a "flat prior" is inherently dependent on arbitrary choices in the way a model is parameterised.

It's not possible for a prior to be "flat" with respect to all re-parameterisations of a continuous parameter in a model. E.g. a flat prior for the variance isn't flat for its inverse (precision) or its square root (the std. dev.), and the choice of which of these alternative parameterisations you use to express the unknown quantity in the model is arbitrary. In the frequentist case it doesn't affect the result of the inference; in the Bayesian case it matters which of the parameterisations you choose your prior to be flat with respect to.