One salient point in this question is that it refers to effect size statistics. The fact that these statistics can be either directional or non-directional, I think is the essence of the answer to the question.
Some effect size statistics are directional †. That is, they can contain positive, negative, or zero values, with a positive value indicating a positive correlation, or, usually, the values for the first group being larger than the second. These include r, Spearman's rho, Kendall's tau, phi for 2 x 2 contingency tables, Cohen's d for the means of two groups, and Cliff's delta for the ranks of two groups.
In these cases, it's a fair interpretation that if the confidence interval does not include zero, then the effect size statistic is "significant". An exception here is that if the statistic is reported as the absolute value ‡ --- that is, always positive --- confidence intervals by bootstrap may not reflect this property. As a final note here, since there are different ways to compute confidence intervals (e.g. by formula or by bootstrap), this conclusion may not always match exactly the related hypothesis test. (That is, say, a 95% CI for Cohen's d may not match exactly the results from a t test).
There are other effect size statistics that are always positive. Examples include r-squared, eta-squared, partial eta-squared, Cramer's v that is used for contingency tables larger than 2 x 2, and the epsilon-squared that is sometimes used for Kruskal-Wallis. Often these are used in situations where it is not easy to convey the effect size in a negative-or-zero-or-positive framework. For example, while we can use phi in a 2 x 2 contingency table, and it makes sense to look at the correlation as positive or negative, in larger tables there may not be a clear way to think about the correlation as positive or negative.
For these statistics, because they are always greater than or equal to zero, an accurate confidence interval would never include values less than zero. Some calculations for confidence intervals may include zero, which would be a potential indicator of "statistically zero effect", but usually calculations won't be so precise. In the case of determining confidence intervals by bootstrap, the range would be unlikely to contain a zero by a method like percentile, although if the confidence interval were determined by normal bootstrap, the interval could cross zero. If this latter method were appropriate for a specific statistic of concern, this approach may perhaps be viable for inference.
One final note, there's an interesting case of Vargha and Delaney's A, which is directional, but where 0.50 indicates no effect.
† I have seen this term used this way for effect size statistics, but I'm not sure it's meaning is universal in this context.
‡ For example, I've seen R functions that report the absolute value of certain effect size statistics.