"giving a point estimate and an associated quantification of its uncertainty is one of the most basic statistical tasks". My argument is that this is only true in a frequentist framing. A bayesian framing would ask, "why do you need a point estimate when you have the posterior?" In the cookie-jar case, what do you actually need to do in the real world that a confidence interval of jars helps you with? Do you win a prize if you guess the jar right? Do you win a prize if you guess the next cookie right? 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? In the last case, a Bayesian directly computes a confidence interval. But in other cases, a confidence interval is just throwing away data for no reason.

Frequentists are fundamentally unable to answer "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?". That question requires a prior; any solution frequentists find is just implicitly assigning a prior to the distribution of jars.