Consider a data generating process with a parameter of interest $\theta$. I would like to estimate $\theta$ as precisely as possible and also quantify the estimation imprecision / uncertainty. I obtain an $n$-size i.i.d. sample from which a confidence interval (CI) for parameter $\theta$ is estimated. The estimator of the CI guarantees that in the long run, 95% of such CIs will include the true parameter value. However, I do not know whether the estimate I got from this particular sample includes the true parameter value or not; nor can I make a statement like the probability that $\theta$ lies in this concrete interval is 95%. Not quite a satisfactory answer for me. I would like to say more about the estimated confidence interval (the particular realization of the respective estimator).
But is this all that the frequentist paradigm has to offer on this question? For example, couldn't I say that I am 95% confident that this particular interval includes the true value, based on the long-run properties of the estimator? (For that matter, I would bet 95 against 5 that the interval actually includes the true value and consider this to to be a fair bet.) Is there a "legitimate" way to express this confidence within the frequentist paradigm, and if so, how should I phrase it?
P.S. Yes, I am aware of Bayesian statistics and the corresponding problem and answer formulations. Nevertheless, this question is intentionally about the frequentist paradigm.