I found explanations of this unsatisfactory as well, including in answers to related questions like this one. Here is a direct explanation in terms of equilibrium of forces.
The green rectangle is the lever, with thickness $h$. (It's upside-down from a typical picture, but that doesn't matter.)
![Diagram of lever internals](https://i.sstatic.net/IAYDQ.png)
We are here reducing the law of the lever to just three simple forces: two stretching (along the top sides of the triangle), and one compressing (along the bottom). See below for an experiment showing this at work in real life.
Geometric proof of why this works
Because the system is in equilibrium, we have $F_x = G_x$ (bottom of the lever resisting the inward push from both sides). Forces $\vec{F}$ and $\vec{G}$ are along the sides of the triangle (pulling between points of contact), and equal to the sum of their horizontal and vertical components:
\begin{equation}
\vec{F} = \vec{F_x} + \vec{F_y} \quad \text{and} \quad \vec{G} = \vec{G_x} + \vec{G_y}
\end{equation}
The rest is by similarity of triangles, looking at the lengths of corresponding vectors:
\begin{equation}
\frac{F_y}{F_x} = \frac{h}{m} \quad \implies \quad m F_y = h F_x \\
\frac{G_y}{G_x} = \frac{h}{n} \quad \implies \quad n G_y = h G_x
\end{equation}
But $F_x = G_x$, since there is no net horizontal force, so in fact
\begin{equation}
m F_y = n G_y
\end{equation}
which is indeed the law of the lever.
It is interesting that thickness $h$ does not affect the result, but is integral to the explanation. I believe this matches the intuition that a rigid rod must have non-zero thickness.
Experimental Setup
Here's an experiment to show that this is indeed sufficient for the law of the lever to work.
![Experimental setup](https://i.sstatic.net/51Ad5.jpg)
The setup is two dumbbells, with weights 10lb and 4lb, hanging at the ends of a metal rod. There is nothing pulling on the middle of the rod; instead two strings are holding its ends. The lengths of the strings are such that the fulcrum is above a point on the rod that divides it in a ratio 4:10. The system is in equilibrium.