Thank you so much for the analysis. Yes that was the logic for the second model.
I guess my (lack) of intuition for the gamma parameters betrayed me. So the summary statistics are not sufficient to condition the individual parameters. Probably because the dispersion information is gone.
I wonder if there are ranges of number of observations/expected value of observations that are less ill-conditioned or if the gamme just stays too flexible to infer one parameter per individual.