For a simple pendulum case, the dynamics is non linear and to study the stability of the system we tend to linearize the system about the equilibrium point ($0$ and $\pi$) and then for stability analysis use lyapunov theorem.
What if we want to do the stability analysis about $30 ^\circ$ and come up with a control to stabilize the system about $30^\circ$? Can we do it? If so, how do we comment on the stability of the system?
I will try to answer your question for a more general class of systems, namely ones that are governed by the following dynamics
$$ \dot{x} = f(x,u) $$
with $x\in\mathbb{R}^n$ and $u\in\mathbb{R}^m$.
You were only considering unactuated equilibrium points, so which satisfy $f(x^*,0)=0$ with $x^*$ an equilibrium point. However you can generalize equilibrium points to pairs $(x^*,u^*)$ such that $f(x^*,u^*)=0$. Linearising around such point, using $\Delta x = x - x^*$ and $\Delta u = u - u^*$, gives
$$ \Delta \dot{x}_\textrm{lin} = f(x^*,u^*) + \underbrace{\frac{\partial f}{\partial x}}_A \Delta x + \underbrace{\frac{\partial f}{\partial u}}_B \Delta u = A\,\Delta x + B\,\Delta u. $$
This can be made locally exponentially stable if $(A,B)$ is stabilizable using a state feedback of the form $\Delta u = -K\,\Delta x$. Or when expressing this in the original coordinates $u = u^* - K(x - x^*)$.