I'm not sure what theta is representing and only skimmed the paper, but especially in social scenarios and across social papers, seems unlikely to assume the same distributions and parameters across tasks & populations. Sometimes comes down to 'is there any effect??' and sometimes a precise notion of effect size in a lucky/clever specific scenario. Likewise, social science is one of the hardest fields to setup a good experiment, and few publications accepts negative results, so mostly only 'good' p-value ranges getting published seems normal. The Wikipedia page on funnel plots shows, afaict, the same criticism of the technique.
Whether about the effect size or how it is reported, funnel plots seem an inappropriate choice for debunking something as general as 'nudges' across heterogeneous studies. Skimming made the metaanalysis feel rather lazy (lack of cross validation, interpretation, ...). Not my field, but I would have had to do some digging before accepting this metaanalysis in review, and by default, would be 'not ready'.