Are the values within a confidence interval those that do not significantly differ from the point estimate? Or, put differently, how do we interpret the values contained in a CI given that the CI is just about the coverage rate? EDIT: Notice that this question is not about the formal equivalence of the t-statistic and the CI (i.e., CI's as significance tests) but how to interpret the values contained in an observed interval.
This is something that I wrote recently for students that may help: A 95% confidence interval of the effect has a 95% probability (in the sense of long-run frequency) of containing the true effect. This probability is a property of the population of intervals that could be computed using the same sampling and measuring procedure. It is not correct, without further assumptions, to state that there is a 95% probability that the true effect lies within the interval. However, if we have only weak prior beliefs about the possible values of the effect, then it is valid, though possibly misleading, to state that there is an approximately 95% probability that the true effect lies in the interval (Greenland and Poole 2013, Gelman, 2013). Perhaps a more useful interpretation is that the interval contains the range of effects that are consistent with the data, in the sense that a t-test would not reject the null hypothesis of a difference between the estimate and any value within the interval (this interpretation does not imply anything about the true value).
Another paper related to the question is https://link.springer.com/article/10.3758/s13423-013-0572-3