Motivation: I was standing in front of a class to introduce into the concept of confidence interval using the example of differences in means (purely frequentist setting) and I was torturing the students with this inconvenient interpretation of confidence intervals. After a while, one student asked: "in the beginning of the lecture you told us that a confidence interval is more useful than a pure p-value because p-values are difficult to interpret and have a different meaning than what many people think they do - but your confidence interval seems to suffer from the same drawback". I immediately thought, "Well, he's got a valid point. Let's go Bayesian then!", but this would have definitely blown the limits of the course.
To cite from AdamO's answer in Interpretation of confidence interval:
The textbook definition of a $100 \times (1-\alpha)$% confidence interval is:
An interval which, under many independent replications of the study under ideal conditions, captures the replicated effect measurement $100 \times (1-\alpha)$% of the time.
Now, when we are in the Bayesian framework, we can construct so-called credible intervals, which allow us to state conclusions like:
[Given the prior distribution that we used,] for a 95% credible interval, the value of interest (e.g. size of treatment effect) lies with a 95% probability in the interval. (source)
Of course, to people who are not trained in frequentist thinking the interpretation of a credible interval is much more intuitive than that of a confidence interval.
I know that one important reason for this difference is that in frequentist framework we do not think of the parameter as a random variable but as a fixed value, while the confidence interval itself is random. However, in certain situations we can prove that the confidence interval calculated from both frameworks will coincide. These "certain situation" particularly cover the most common cases, i.e. the case of mean differences and linear regression with weakly informative prior distribtions.
Here comes my question: if I can show that Bayesian credible intervals and frequentist confidence intervals coincide why can't I interpret the frequentist confidence interval in the Bayesian way?