Intuitively why time period of shm do not depend on displacement from mean position, ie. amplitude

Question

Why simple harmonic motion time period equation is independent from displacement from mean position?

Answer 1

The expression for time period does not contain displacement in it and that is why it does not depend on displacement. But, that however is not what you are looking for I think. So, think about the the speed of the bob. It is variable. The further you displace it, more the potential energy it acquires and hence collects more speed when coming back towards mean position.

$$mgh = \frac{mv^2}{2}$$

Here, $h$ is the vertical displacement form mean position.

So, as the displacement gets bigger, so does the velocity. Thus the time period remains constant.