While I sympathize with your unease about making time a privileged coordinate, even in conventional quantum mechanics an operator for time seems difficult. What

While I sympathize with your unease about making time a privileged coordinate, even in conventional quantum mechanics an operator for time seems difficult. What would that operator measure? The time at which a given object "is"? The whole point of physics is to describe the dynamic nature of reality, parametrized by time. Speaking of which, time(-of-arrival) measurements in quantum mechanics have recently attracted quite some interest: The classic Copenhagen formalism doesn't seem to give an answer here (or at least not a unique one – it depends on how you perform the calculation). Meanwhile, Bohmian mechanics does seem to make a precise prediction. It will be interesting to see what experiments will yield.