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You are entirely right, whether something that depends on higher-order derivatives can be called Newtonian is debatable. Personally I don’t really care either way, as this is just a label. Newton did not mention higher-order derivatives but on the other hand they are a trivial extension to the mathematical framework. It is difficult to call a body at rest if any of the derivatives of the position is not zero, because then it will start moving instantaneously so it is hard to read the first law otherwise. And the second law does not care about anything other than acceleration. And there certainly isn’t anything that prevents us from using clever shape to roll balls on, as long as the shape make physical sense.

What this does not change, however, is that the dome does not demonstrate non-determinism. The apparent demonstration hinges on logical errors that remain errors regardless of the framework used, be it classical or quantum mechanics, or relativity.