What reasons are there for describing the harmonic oscillator as being so important in physics?

The harmonic oscillator tends to show up when you're expanding a potential function around non-degenerate critical points.

The simplest example is a physical system described by a map $t \mapsto \phi(t) \in \mathbb{R}$. If the energy function for this system has the form $E(\phi) = \frac{1}{2}|\dot{\phi}|^2 + V(\phi)$, with $V$ bounded below, then the lowest energy states are going to be of the form $\phi_0(t) = \phi_0$, where the constant $\phi_0$ is a minimum of $V$, hence a critical point. So, if your map $\phi$ never deviates too much from $\phi_0$ and $\phi_0$ is a non-degenerage critical point, you can approximate the energy function by $E(\phi) = |\frac{1}{2}\dot{\phi}|^2 + \frac{1}{2}V''(\phi_0)(\phi-\phi_0)^2$.

In other words, the harmonic oscillator potential describes small disturbances around "generic" minima of an energy function. This situation comes up all the time in physics. For example: it shows up in Witten's Supersymmetry & Morse Theory paper, which I think would have been well-known to people working on topology and analysis in the 1980s.

What reasons are there for describing the harmonic oscillator as being so important in physics?

The harmonic oscillator tends to show up when you're expanding a potential function around non-degenerate critical points.

The simplest example is a physical system described by a map $t \mapsto \phi(t) \in \mathbb{R}$. If the energy function for this system has the form $E(\phi) = \frac{1}{2}|\dot{\phi}|^2 + V(\phi)$, with $V$ bounded below, then the lowest energy states are going to be of the form $\phi_0(t) = \phi_0$, where the constant $\phi_0$ is a minimum of $V$, hence a critical point. So, if your map $\phi$ never deviates too much from $\phi_0$ and $\phi_0$ is a non-degenerage critical point, you can approximate the energy function by $E(\phi) = |\frac{1}{2}\dot{\phi}|^2 + V(\phi_0)+\frac{1}{2}V''(\phi_0)(\phi-\phi_0)^2$.

In other words, the harmonic oscillator potential describes small disturbances around "generic" minima of an energy function. This situation comes up all the time in physics. For example: it shows up in Witten's Supersymmetry & Morse Theory paper, which I think would have been well-known to people working on topology and analysis in the 1980s.