What the evidence does and does not say
The weekly review is the core ritual of Getting Things Done, and it is worth being straight about its status: there is no randomized controlled trial of GTD. The method has never been tested as a package. Anyone telling you it is scientifically proven is overselling.
What does have support is the mechanism the review runs on. Masicampo and Baumeister (2011) showed that unfinished tasks intrude on unrelated thinking, degrading performance on things that have nothing to do with them, and that the intrusion goes away when you make a specific plan for the unfinished task. Not when you finish it. When you plan it. That is a fairly direct experimental case for the practice of sitting down once a week and converting every vague obligation into a concrete next action.
The second supported piece is planning itself. Gollwitzer and Sheeran's meta-analysis of 94 studies found that if-then plans naming a specific cue and action improved goal attainment with a medium-to-large effect (d = 0.65). A weekly review that produces sentences like "Wednesday after standup, I draft the vendor email" is manufacturing those plans in bulk. A review that produces a list of topics is not.
Keep expectations calibrated. The largest time management meta-analysis (Aeon, Faber and Panaccio, 2021) found a moderate link to performance and a stronger one to wellbeing and reduced stress. A weekly review is more likely to make your week feel under control than to make you dramatically more productive, and that is a legitimate thing to want.