Meta-analyzing incidence (count) data with many zero cells I am conducting a meta-analysis on the incidence of a rare event. I have 15 single-arm studies with events occurring in only 2 studies.
The metarate function in the meta package allows for meta-analysis of incidence, but the only option I can find for managing zero-count data is a fixed increase in zero data. I would rather use a more-evidence based approach such an alternative continuity correction.
Is there a way to do alternative continuity correction with metarate? I am open to other evidence-based methods and other package solutions.

PS I considered logistic regression as there are many strategies for zero-count data (zero inflated, negative binomial, and hurdle models) but I am dissuaded by the fact that I have no important covariates because this analysis is at the level of the study, not the patient.

I realize this is at the study-level and your first column looks like the number of subjects in each study.  Do you have another column that contains subject-years for each study?  This will be helpful for calculating the exposure-adjusted mean since it is likely not every subject was followed for the same amount of time.
My suggestion would be to feed this into a glm package.  The regression could be an intercept only.  You do not necessarily need to include covariates.  Perhaps you could use a Poisson model with an additional dispersion parameter to account for under- or over-dispersion.  This avoids model convergence issues that often occur with the negative binomial distribution and you do not need to specify a separate model for excess zero counts.
When performing the meta-analysis you can assume the true mean rates being investigated across all the studies are equivalent and pool the standard errors.  Alternatively, you can assume the true mean rates are different across or all of the studies and analyze those separately.  This is analogous to determining whether to adjust an outcome model using a covariate with several factors.  This can be investigated by adjusting for study as a fixed effect and examining the type III test.
Many authors and practitioners will suggest a random effects meta-analysis to account for heterogeneity in parameters, but I recommend against this since the population parameters aren't actually sampled from a population of populations.  Heterogeneity alone does not constitute a random variable.  It is the act of sampling that constitutes a random variable.  Here is a related paper on meta-analysis.
