A generalization which applies to levers, pulleys, and hydraulics is mechanical advantage while conserving energy. You have a system with input work and output work (energy) that are the same, ignoring frictional losses.
Recall that work is force over distance. The mechanical system relates the input and output distances by a scalar coefficient. Since the working distances are related by a ratio, the working forces are related by the reciprocal of that ratio.
You can find the lever and fulcrum ratio with simple geometry. The input and output lever segments are radii, and the travel is distance along two arcs. Since the arc length is directly proportional to radius, the ratio of lever radii translates directly to the same ratio of arc lengths, and the reciprocal ratio is the force multiplier. Your 10:1 lever sweeps 10:1 arc lengths and balances with 1:10 opposing forces.
Yes, that's an excellent point, but I think the lever law is more general than that. For example, it continues to apply when the lever in question is stationary, even though no value of the forces involved would violate conservation. In fact, it holds to higher precision in that situation because your measurements aren't confounded by vibration and accelerating masses.
Maybe you can derive it from some kind of generalization of Hooke’s Law to cover nonlinear stress–strain relationships, elastic hysteresis, anisotropy, viscoelastic behavior, and so on, but it's not obvious to me what that would be. Also, I feel like the concept of angular moments acting to produce angular acceleration is simpler and more general than all that stuff, but I'm not sure if conservation of energy and geometry alone are sufficient to derive it.
Recall that work is force over distance. The mechanical system relates the input and output distances by a scalar coefficient. Since the working distances are related by a ratio, the working forces are related by the reciprocal of that ratio.
You can find the lever and fulcrum ratio with simple geometry. The input and output lever segments are radii, and the travel is distance along two arcs. Since the arc length is directly proportional to radius, the ratio of lever radii translates directly to the same ratio of arc lengths, and the reciprocal ratio is the force multiplier. Your 10:1 lever sweeps 10:1 arc lengths and balances with 1:10 opposing forces.