Well, I am not really interested in cookie jars. But I am interested in, for example, particle physics. There we need need simple ways to communicate point estimates and the associated uncertainties for various parameters of nature. Intervals are a convenient way to do this. Frequentist confidence intervals have the virtue that they will cover the true parameter at the nominal rate. Bayesian credible intervals in general have no such guarantee. In many cases we can find Bayesian-inspired interval estimators that have good coverage properties. But in some cases there is an irreconcilable conflict. And there you have to choose between long-run correctness and being Bayesian.

You might claim that we should do away with intervals and just report posteriors for all physical quantities. This complicates matters slightly without solving the problem. If the true parameters, when ultimately known, consistently end up in very low density regions of the probability distribution (such as far in the tails), we would regard our uncertainty estimates as poor. Again, it is not hard to construct examples where Bayesian methods have poor coverage properties in this sense.

(Also, a minor point: Regarding "Do you need to select a strategy up front which gives at least a 70% chance of being right about the jar over iterations where the jar is fixed but the data varies?", one does not need to fix the jar for frequentist methods to have good guarantees. See Wasserman's simulation with the median.)

Re: "Given that I drew a cookie with 2 chips on the first draw, what is the chance I draw a cookie with 0 chips on my second draw?", this is not a question about estimating an unknown parameter of a distribution, so it's not statistical in the sense Wasserman is talking about. It's just an elementary probability question that requires knowing something about the jars to answer. Both frequentist and Bayesian statisticians agree on the validity of Bayes rule (and hence how to answer this question once the relevant information is known or assumed); where they differ is on how to conceptualize and estimate unknown parameters of probability distributions.