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.
If this seems a bit odd (and it did to me at first!) think about it this way:
Bayesian methods work by averaging over a bunch of different models / different values of the parameters.
What it means to compute a mean depends on the parameterisation in which you do it: simplest example being that an arithmetic mean is not in general the same as a geometric mean, or a harmonic mean.
There's no "neutral" / parameterisation-independent way to specify how this averaging is done, so if you care about the average case, you're going to have commit to doing it some particular favoured parameterisation. Choosing that parameterisation is equivalent to choosing the prior.
Frequentist methods avoid the need for this decision; the price they pay is that without a prior they're unable to condition on the observed data. They must consider every parameter value and its resulting sampling distribution separately and can't average over them.
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.