Does one still need to report p-value & effect size if one has reported the confidence interval (CI)?

I am unclear about the degree of overlap in the information contained by the CI as opposed to that contained in the p-value and effect size of that same statistic/estimate.

I know of course that if a 95% CI does not include the zero value of a particular effect, then it can be inferred that said effect is statistically significant, in the same way that this would be inferred from a p<0.05. And, of course, that CIs can be calculated from p-values, and vice-versa.

But what is unclear to me is:

1) if the CI is reported, is there any need at all to still provide the p-value? In other words, does it give any extra information that the CI does not already give?

2) if I want to describe, say, the difference in means between two groups, does it still make sense to provide a measure of effect size such as Cohen's d if I have already provided the CI of the sample point-estimate?

Reading the paper "Confidence Interval or P-Value?" by du Prel et al. (2009) has left me still in doubt unfortunately, so further reading that addresses my question would be appreciated in an answer - thanks in advance.

I should also mentin that this question follows on from an earlier one to which I never obtained a satisfactory answer, just as searching previous questions on here was not entirely enlightening.

While information such as differences in means and p-values can often be back-calculated from a confidence interval (especially if it is of the form $\hat{\delta}\pm c_{1-\alpha/2} \text{SE}(\hat{\delta})$) potentially after some transformation), it is not easy for people to do this in their head. So, it is better to do these calculatons for them. Particularly the difference in means should really be provided in my opinion, while it is in many fields expected to also be provided p-values (even if people usually mis-interpret them).
I will share what I tell my students: our job in conveying statistical information to our "clients" is to quantify what was observed and to provide some measure of how this evidence can be assessed. I would argue that the sample statistics and effect sizes address the former, and confidence intervals (CI) or $P$-values address the latter. Thus, I would suggest reporting both the CI and the effect size, such as $d$ or $R^2$. However, reporting both the CI and the $P$-value would be redundant.
Regarding the 2nd query, the size of the CI depends on $n$, and thus can be misleading. That is, you can have two CIs with the same distance from zero and the same $d$. One CI could be very large and the other very small. Thus, you do provide additional information by reporting the effect size, as well.