[kgwgk]

It’s not true that “there are always priors”. There are no priors when you calculate the area of a triangle, because priors are not a thing in geometry. Priors are not a thing in frequentist inference either.

You may do a Bayesian calculation that looks similar to a frequentist calculation but it will be conceptually different. The result is not really comparable: a frequentist confidence interval and a Bayesian credible interval are completely different things even if the numerical values of the limits coincide.

[zozbot234]

Frequentist confidence intervals as generally interpreted are not even compatible with the likelihood principle. There's really not much of a proper foundation for that interpretation of the "numerical values".

[kgwgk]

What does “as generally interpreted” mean? There is one valid way to interpret confidence intervals. The point is that it’s not based on a posterior probability and there is no prior probability there either.