Turning on Large Deflection causes the solution to take more time, that is why it is off by default, allowing linear models to solve in the minimum time.
In the fishing rod example, the material remains linear, so on a local scale, Hooke's Law is valid, but at the overall level of the strucure, Hooke's law applied to the rod as a whole is violated because the force-deflection curve does not remain linear at a large deflection.
Any time Plasticity is included in the model so that the material will respond differently after the yield stress is reached should have Large Deflection turned on.