What one should not say when using frequentist inference is, "There is 95% probability that the unknown fixed true theta is within the computed confidence interval." To the frequentist probability describes the emergent pattern over many (observable!) samples and is not a statement about a single event. However, understanding the long-run emergent pattern gives us confidence in what to expect in a single event. The key is to replace "probability" with "confidence," i.e. "I am 95% confident that the unknown fixed true theta is within the computed confidence interval."
This is analogous to knowing the bias of a coin is 0.95 in favor of heads (95% of the time the coin lands heads) and the confidence this knowledge of the long-run proportion imbues regarding the outcome of a single flip. If asked how confident you are that the coin will land heads (or has already landed heads), you would say you are 95% confident based on its long-run performance.
To the frequentist, the limiting proportion is the probability and our confidence is a result of knowing this limiting proportion. To the Bayesian, the long-run emergent pattern over many samples is not a probability. The belief of the experimenter is the probability. The Bayesian is also willing to make (belief) probability statements about an unobservable population parameter without any connection to sampling. Such statements are not verifiable statements about the actual parameter, the hypothesis, nor the experiment. These are statements about the experimenter. The frequentist is not willing to make such statements.
Here is a related thread showing the interpretation of frequentist confidence and Bayesian belief in the context of a COVID screening test. Here is a related thread comparing frequentist and Bayesian inference for a binomial proportion near 0 or 1. To the frequentist, the Bayesian posterior can be viewed as a crude approximate p-value function showing p-values and confidence intervals of all levels.