A lot of this is missing the point. Norton's dome shows that the mathematical formalism of Newtonian Mechanics admits non-deterministic solutions. That's all. It's only an approximation in the real world after all, so non-physical solutions aren't of real interest in that context, but they are in a mathematical one.
The fact that the author quibbles about units, when they've failed to notice the Norton took `g=1`, makes it seem as if they're not familiar with conventions in mathematical physics. The addition of a constant `k` adds no value other than converting between units, which aren't of interest here other than as a bookkeeping mechanism, because we're not dealing with any numerical quantities.