One caveat here is that the larger the sample size the less a low p-value implies a large effect size, and I don’t feel like I have a good intuition about how exactly the two are connected.
An alternative to p-value NHST that’s still frequentist is to produce the point estimate and confidence intervals for the effect size of interest. Assuming they are defined the same way (one-side vs two-sided, etc.), they will give the same conclusion if you choose to interpret them as a dichotomous ‘significant or not’ status. (I.e., the 95% CI will exclude the null in the same cases as where the p-value will be below 0.05.) But with the CIs you’re getting that indication of the size of effect that you were craving.
An alternative to p-value NHST that’s still frequentist is to produce the point estimate and confidence intervals for the effect size of interest. Assuming they are defined the same way (one-side vs two-sided, etc.), they will give the same conclusion if you choose to interpret them as a dichotomous ‘significant or not’ status. (I.e., the 95% CI will exclude the null in the same cases as where the p-value will be below 0.05.) But with the CIs you’re getting that indication of the size of effect that you were craving.