I agree that often in practice a confidence interval will end up being similar to a Bayesian credible interval. However, having a flat prior is not enough to guarantee this: the post gives an example with a uniform prior and a valid 70% confidence set that only captures 37.5% of posterior mass, and even an example with a uniform prior and a valid 70% confidence interval that captures 0% of posterior mass (because, similarly to your example, the set is empty).