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The advantage is that [itex]\omega = d\theta/d\tau[/itex] is what the experimenter measures directly.It isn't that hard to understand. The coordinate transformations are easy in terms of coordinate time, they would be much more difficult in terms of proper time. In fact, with your alternate definition of [itex]\omega = d\theta/d\tau[/itex], what exactly are the transformations between your coordinates and an inertial coordinate system? And, what is the metric in your coordinate system?

Btw, the advantage of my approach is that it applies for any arbitrary worldline in any arbitrary coordinate system and will always give the correct proper acceleration. A derivation based on potentials only works for static spacetimes where potentials can be defined, and I am not sure that it works in any coordinates where the particle is not stationary. I don't see any advantage to it when the general approach is so straightforward.

As to the potentials, they can always be calculated, see Moller, see Gron.