Norton's dome is not a valid paradox, it just exploits in an overly complicated way the fact that almost no handbook of physics bothers to present a complete set of axioms for the classical mechanics (and even less for relativistic or quantum mechanics). So the "paradox" is based on the sloppy teaching of physics from a mathematical point of view.
The form of the Norton's dome does not matter. The so-called paradox is just a random example of the fact that there exist multiple functions of a variable that have in the origin the same values for the function and for the first 2 derivatives, e.g. various pairs of polynomials of the 4th order.
Therefore if you accept any function of time as describing a possible motion, you can always find motions that at some moment in time have the same position, velocity and acceleration.
This is not an example of indeterminacy in classic mechanics, because one of the axioms of the classic mechanics is that all the forces that exist in nature are such that the state of a mechanical system is completely determined by the positions, velocities and accelerations of its components (in other words, a mechanical system must be described by a system of differential equations of the second degree that has a unique solution).
There is no difficulty of imagining other kinds of forces, for which this assumption is not true, but a theory where such forces exist is no longer the Newtonian mechanics, in the same way as any geometry where Euclid's axiom of parallels is not true is no longer an Euclidean geometry.
If Newtonian mechanics were a correct model for the World, a ball would remain forever on the top of the dome, without ever falling. In reality, even assuming the validity of Newtonian mechanics, the main reason why any attempt to test this experimentally would fail is the thermal motion, due to which a ball can never be at rest, so it would always start immediately to fall in a random direction.
The violation of the axioms is why the so-called different solutions are not solutions within Newtonian mechanics.
On the other hand the argument that the initial state could be obtained by launching the ball towards the top, and then time reversal would demonstrate a valid solution, it is also wrong, because the so-called solution cannot be obtained by time reversal.
If the ball is launched with only enough energy to reach the top, so it will come to rest, then it requires an infinite time to reach the top. Reversing the time means that the ball will remain on the top for an infinite time, without falling, as expected.