First, there is not such thing as a ‘uninformative’ prior; it’s a misnomer. They can change drastically based on your paramerization (cf change of variables in integration).

Second, I think the nod to robust methods is what’s often called regularization in frequentist statistics. There are cases where regularization and priors lead to the same methodology (cf L1 regularized fits and exponential priors) but the interpretation of the results is different. Bayesian claim they get stronger results but that’s because they make what are ultimately unjustified assumptions. My point is that if they were fully justified, they have to use frequentist methods.

One standard way to get uninformative priors is to make them invariant under the transformation groups which are relevant given the symmetries in the problem.