You're using correlation in what I would consider a weird way. Randomization is intended to control for selection effects to reduce confounds, but when somebody says correlational study I get in my head that they mean an observational study in which no attempt was made to determine predictive causation. When an effect shows up in a nonrandomized study, it's not that you can't determine whether the effect was causative; it's that it's more difficult to determine whether the causation was due to the independent variable or an extraneous variable unrelated to the independent variable. It's not a question of whether the effect is due to correlation or causation, but whether the relationship between the independent and dependent variable even exists at all.