Yes it's amazing that it can find an equation that fits the pendulum's motion. But if you didn't already know the physics math, it would just seem weird and arbitrary.
No more weird and arbitrary than the motion of the pendulum itself. I think you're underestimating the human intelligence component of the system. Include in the dataset an additional variable for rope length, record the data for multiple lengths - there, you've just isolated a variable in the equation and it is no longer arbitrary. The scientific method still works, but now you're working on a much more tightly bounded problem.