$$\oint \frac{dz}z = 2\pi i$$

I've seen the derivation of it using the parametrisation. Since this result is used all the time in my complex analysis course, i'd like to understand this intuïtively. I tried making sense of it like this:

$$\oint \frac{dz}{z} = \mathrm{mean}\left(\frac{1}{z}\right) \cdot \oint dz = \mathrm{mean}\left(\frac 1 z\right)\cdot 2\pi$$

Or by thinking of it like a 2D version of the flux integral of a field through a sphere. Then the singularity in the origin would act like a source of field lines.

Is there another way to think about this?