For all shapes other than Norton's dome, it takes a ball infinite time to come to rest at the top. Therefore the time reversed path starts infintly long ago, not at an arbitrary time. This removes the indeterminacy from the solution.
Or so it seems. There is still nothing needed to make the motion start. Many paths rolling up tthe cone from different sides lead to the same state in the infinite future. Time reversal can lead to any of them.
This still does not show the universe is indetermanistic. This uses classical Newtonian mechanics and perfectly rigid objects on infinitely precise trajectories. These are approximations and idealizations. Nondetermanism is only found in corner cases where the approximations do not match the universe well. Of course, quantum mechanics does show the universe is nondetermanistic.