This is a very interesting discussion - Hope you dont mind I made it into a new post.
Before I test your model, a quick comment:
I am not sure it is fair to say Metropolis is performing better or even correct. The posterior is multimode and Metropolis is basically stuck in one mode - that’s also why initializing at omega = 3.5 for NUTS also produce nice trace. “Stuck in a mode” might appear to be stationarity but it is actually not inferencing the model correctly.