Newton second law is known as fundamental law of mechanics, because it is supposed to solve the fundamental problem of mechanics, that is, finding the position of a particle at any given moment in time, i.e., to find
$$
x=f(t)
$$
Plot of $x=f(t)$ can only be a straight line (a special type of curve with curvature=0) or a curve (any curve with curvature <>0). However, any plot just depends on a starting point (current value or initial value) and on how it changes when moving forward or backwards from from that given point. Such a change is represented by curve’s curvature, i.e, from this point on, your choices are, continue straight ahead (curvature=0), go up (curvature>0) or go down (curvature<0). How much you go down or up, dependes on curvature’s magnitude.
It happens that curvature depends only on second and first order derivatives
$$
\textrm{curvature}=\frac{x^{\prime\prime}}{\left(1+{x^\prime}^2\right)^{3/2}}
$$
So, any possible curve for $x=f(t)$ would just be characterized by its first and second derivatives provided that force in $F(x’,x,t)=m \frac{d^2x}{dt^2}$is properly defined.
In an universe with higher order derivative (with respect to us), one could always set that universe’s straight line to be the solution of our $n^{th}-1$ derivative, meaning that in that particular universe Newton’s first law would be a curve with respect to us, but not with respect to themselves, and all they would need to discriminate their natural state of motion (motion in a systems straight line) would be that universe’s second order derivative.
In summary second order derivative is all one needs to differentiate natural states of motion with affected states of motion.
One needs to understand that even though many quantities like, energy, momentum, velocity, acceleration, force, jerk and so on… are (and may be) defined in mechanics being afterwords useful in other branches of science, the ultimate goal of mechanics is to find $x=f(t)$.