Both the authors of this study and meta-analysts generally have followup responses and other diagnostics and models that work under degraded assumptions, so I don't think it makes a ton of sense to respond to "this is the absolute most introductory example of why this kind of intuition works" with "it doesn't work in <XYZ> case". There are a variety of ways to think through what data looks like under publication bias or not irrespective of what the "correct" effect would be.

I'd also add that this is a paper responding to an existing meta-analysis, so the claim that it's impossible to get a quantity of interest for a meta-analysis because the constituent studies are being inappropriately aggregated is itself an argument in favour of this paper's rejection of the original paper's finding.

I personally just rejected an m-a at a social science journal on thursday because I thought it suffered from unaddressed garbage-in garbage-out along the lines you mention (non-experimental data, no attempt to pin down causal identification, inadequate qualitative discussion of risks of publication bias, unclear QOI), so I am sympathetic to the criticisms, but just know that there are next steps. :)