I have learned that the covariant derive is just the normal derive minus the normal component, 9:53 of this video.
When our space is an intrinsic plane, then the covariant derivative just becomes the normal derivative since in an intrinsic plane since there is no longer a 3rd dimension. The covariant derive of the metric tensor in an intrinsic plane would just be the normal derivative (i.e the rate of change of that metric tensor).
So, since the metric tensor changes across space for a plane that is intrinsically curved, why is the rate of change of that metric tensor (the covariant derivative) zero? An explanation I have heard is that it’s just constrained to be zero by choosing a specific connection. I disagree with this explanation because we get the connection that we are dealing with by just taking the normal derivative of a basis vector, so it is a property of space (refer 17:37 of this video).
I know the proof why it’s zero mathematically, but how do I reason this intuitively the metric tensor is a property that changes from point to point in space?