The explanation is quite simple.
1) The big lever is used to "store energy"
2) When the pendulum is in the lowest point, the lever holding pendulum pivot will drop by height H, thus accepting mechanical energy equal to H x (F + G), where G is the weight of the pendulum (gravitational force = m x g) and F is the centrifugal force. If this drop occurs in the lowest point, it does not affect the movement of the pendulum, because the respective acceleration is perpendicular to the movement of the pendulum (it cannot hang on a wire, the connection between pendulum and its pivot must be stiff). The drop decreases the POTENTIAL energy of the whole pendulum system (pivot and pendulum) and stores it as increase of POTENTIAL energy of the opposite side of the lever.
3) When the pendulum is rising, the big lever comes out of balance, because the vertical force on the pendulum pivot decreases. The pivot is raised and pendulum kets a "kick", but at this time parallel or almost parallel to the upward movement. This kick increases KINETIC energy of the pendulum. This energy is taken from decrease in POTENTIAL energy of the lever. Previously stored POTENTIAL energy is transformed to KINETIC energy.
4) Where does the energy increase come from? Well, the POTENTIAL energy in this system obviously comes from gravitational force. In full accord with Newton's third law, when the whole system "pumps" potential energy, there is an opposite movement of the other body that causes the gravitational force. Which is the Earth... Thus, every "drop" is accompanied by a "raise" of the Earth, but that is of course immeasurable quantity due to the huge ratio between mass of the Earth and mass of the pendulum-lever system. It is similar when you launch a rocket or jump on your feet. Your feet push the Earth away a little bit, but this little bit is negligible compared to the height of the jump.
So, the guy who said that Two-Stage Mechanical Oscillator is a claimed "perpetuum mobile" should go back to Junior High or whatever is their country's basic education institution and let some teacher explain them again the Newton's three laws of motion. In energy analysis you cannot just deliberately neglect one element of the whole system. If you do that, your result will be nonsense.
Taking $F_{in}$ as the reference, it is $F_{in}= F_{out} \sin(\theta)$, which makes mechanical advantage proportional to $\frac{1}{\sin(\theta)}$, or $\frac{l}{w}$ for small $θ$.
Taking $F_{out}$ as the reference, as you have chosen, it is $F_{out}= F_{in} \cos(θ)$, which makes mechanical advantage proportional to $\frac{1}{\cos(θ)}$, or $\frac{1}{(1-sin^2(θ))}$ where a simple approximation is not so clear.
Best Answer
The length of the lever arm is the same regardless of curvature of the arm between the pivot and the point at which force is applied. However, if force is applied perpendicularly to the surface of the grip on the lever, the tilt of the grip relative to the line between the point where force is applied and the pivot can make a difference: the applied torque will greatest if the force is applied directly perpendicular to that line.