Ok so the way I see it, let's strip away the physics first. Then we're left with the math. So the question you're posing is, given only $\frac{d^nx(t)}{dt^n}|_{t=c} \forall n$, can we reconstruct the function $x(t)$ uniquely in some domain $(a,b)$ s.t. $c \in (a,b)$?
Well the answer is no. Even if you impose a $C^\infty$ condition on $x(t)$, the example you gave (with $x(0) = 0$) shows that it is not possible. Well, a necessary and sufficient condition requires complex analysis: all holomorphic functions in $(a,b)$ can be reconstructed just based on knowledge of all its derivatives at one point. Note that holomorphicity is equivalent to analyticity.
Thus, we need the extra a priori condition that our function is holomorphic in $(a,b)$, in order to determine it uniquely.
Now let's turn on the physics and try to interpret the question. Here I'm just offering my opinions/speculating, so feel free to correct me or post comments.
I'll first like to point out that this is a purely kinematical question, and not a dynamical one. In other words, this question makes no reference to Newton's laws at all. So this question is actually somewhat different from the Norton's dome paradox, in which the question there is whether Newton's law $F=ma$ should be deterministic or not, since the equation of motion was derived specifically from $F = ma$.
Now why do we think the Norton dome paradox is a problem? Because somehow we inherently believe that Newton's laws should be deterministic - given an initial condition that evolves due to Newton's laws, we should get a unique flow in phase space. But obviously this is only guaranteed if the force equation obeys certain conditions like Lipschitz continuity. The resolution to this problem is either that we arbitrarily impose a Lipschitz continuity condition on the force, or that we say that Newton's theory is not completely correct. Since Norton's dome gives us an actual scenario in which the force is not Lipschitz-continuous, we are forced to take the second view, and the better theory that supersedes Newtonian theory is of course, QM.
Now for this question, it doesn't actually say anything about Newtonian classical mechanics, and so it cannot give any 'insight' about the non-determinism of Newtonian mechanics, unlike the Norton dome problem. While it might seem un-intuitive that given the position, velocity, acceleration, jerk, etc. of a particle at one in time, we do not know its position at all times, I would like to argue that this is purely a failure in our understanding of mathematics, as opposed to a shortcoming of Newtonian mechanics.
Thus the point I'm trying to make is that kinematics is not the same as dynamics, although the two are intimately intertwined. This might be a bold (but true) statement, but kinematics is not physics - it is just math littered with physical terms: Position, velocity, acceleration can be replaced with function, first derivative of function, second derivative of function; 'distance traveled in $t$ seconds' is just 'integral of $x'(t)$' etc. So whatever paradoxes we have about kinematics ultimately stem from misunderstandings about mathematics.