Why is the simple harmonic motion idealization inaccurate?

The actual restoring force in a simple pendulum is not proportional to the angle, but to the sine of the angle (i.e. angular acceleration is equal to $-\frac{g\sin(\theta)}{l}$, not $-\frac{g~\theta}{l}$ ). The actual solution to the differential equation for the pendulum is

$$\theta (t)= 2\ \mathrm{am}\left(\frac{\sqrt{2 g+l c_1} \left(t+c_2\right)}{2 \sqrt{l}}\bigg|\frac{4g}{2 g+l c_1}\right)$$

Where $c_1$ is the initial angular velocity and $c_2$ is the initial angle. The term following the vertical line is the parameter of the Jacobi amplitude function $\mathrm{am}$, which is a kind of elliptic integral.

This is quite different from the customary simplified solution

$$\theta(t)=c_1\cos\left(\sqrt{\frac{g}{l}}t+\delta\right)$$

The small angle approximation is only valid to a first order approximation (by Taylor expansion $\sin(\theta)=\theta-\frac{\theta^3}{3!} + O(\theta^5)$).

And Hooke's Law itself is inaccurate for large displacements of a spring, which can cause the spring to break or bend.

The problem here isn't so much calculus as the assumptions that are made about the system. The solution can only be as accurate as the assumptions made, no matter how accurate the solutions of the equations are.

For a pendulum where the distance between the centre of gravity and the anchor point is $l$ and the mass is $m$, the equation of motion is:

$$\frac{d^2\theta}{dt^2}+\frac{g}{l}\sin \theta=0$$

Where $\theta$ is the angle between the pendulum and the vertical.

But this is mathematically difficult to solve, so we invoke the small angle approximation:

$$\sin \theta \approx \theta$$

This makes the equation easy to solve but its solutions only approximate.

Similarly for spring mass systems we usually assume the spring to be Hookean but for many real world systems that's only an approximation.

Often other assumptions about an oscillating system, such as no friction/no drag will also introduce further inaccuracy of the model vis-a-vis reality.

For a pendulum, you use the approximation $\sin(\theta)\approx\theta$ in the derivation of the simple harmonic equation of motion, which is only valid for small angles.

For a spring, it is Hooke's law itself that is only valid for reasonably small streching of the spring - a spring to which you apply hundred times the force needed to strech it by 10% usually will not be ten times its length, if it is still in one piece at all.