I might be glancing over the obvious issue, but I don’t think you have enough information to constrain your hyperparameters alpha and beta. You have one single theta with which they have to agree, so I would expect their posteriors to not be that much different from the priors.
Similarly your theta posterior would be much more uncertain if you had a single observation in the binomial (that is n=1).
Remember that the hyperpriors are conditionally independent from the likelihood, once we consider theta.