How exactly did you do your trials?

Somethings to think about: Does the actual parameter value that you use in your simulation make a difference on which method is preferred? Can you think of a way to analyze the difference between the Bayesian estimate and the frequentist method without necessarily having to do simulations? Which prior did you use? Does this make a difference?

As for your question of whether or not Bayesian methods are better - it highly depends on what you mean by "better". The questions I ask in the previous paragraph hopefully shed some light on that and make you think about it a little more.

You say you don't want to compare them on a philosophical basis and that's alright (hopefully I gave you some ideas on how to compare them on a more quantitative basis) but personally the philosophical basis is the reason I prefer Bayesian methods.