In all fairness, the reasons why meta-analyses are problematic are statistical in nature: As long as the data is not collected with the very same scope and method, it cannot be precluded that different moderating effects, especially in choosing the participants and asking the questions, are at work. Simply hoping for the best and that they kind of even out is not exactly scientific, see e.g. publication bias.
My own statistics lecturer pointed out that after-the-fact analyses are something that differs from scientific inquiry through statistical methods. Testing is completely depending on the hypothesis you test and includes every single phase.
I have seen meta-analyses that completely ignored important factors like different takes on what the population is, different metabolic peculiarities because of ethnicity and culture (in medicine papers) etc, etc. The problem is not "saying something about different papers", the problem is "pretending you have one single set of data while you, in fact, have dozens of different ones where you cannot even determine the differences and commonalities".
For literature on how to circumvent all these objections (which surely is not philosophical), see: Schmidt, F. L., & Hunter, J. E. (2014). Methods of meta-analysis: Correcting error and bias in research findings. Sage publications.
That being said: Although there is a huge discussion going on in the philosophy of science about what it takes to provide sufficient scientific validity and reliability (esp. regarding causal connections), the mathematical or statistical method itself is seldom the point of the particularly philosophical contribution to the discussion. Philosophers do, as far as I am aware of, refer to the standards established in the sciences and statistics.
Overall, the question in Health.SE seems to be a good fit and is considered a good fit as far as I can see. Maybe a carefully phrased question about the statistical obstacles in meta-analyses could be a good fit for Mathematics.SE as well, but it would have to be very technical.
As @Keelan points out in his comment, CrossValidated.SE would probably be the perfect place, as both the purely technical as well as problems in application of statistical methods are in the scope of that community.