Why does the circle naturally divide into $6$?

I think you are basically asking: Why do exactly $6$ equilateral triangles fit in a circle? and pointing out different instances of it.

In which case: it's because the angles of a triangle sum to $180°$, which is "half a circle".

Since an equilateral triangle has all three angles equal, each angle has to be a third of $180°$—or a "sixth of a circle".

So exactly $6$ of them fit around the circle's centre.

Construct the first two circles and connect their centers. Notice that the length of this connection is the radius (which we'll define as a unit). Now connect the centers to one of the intersections and note that you have constructed an equilateral triangle (with angles each $\frac \pi 3)$.

Six of these triangles would thus exactly fit in a revolution of $2\pi$.

(Another way of thinking of it is to notice that the length of the arc subtended by the angle at a unit circle center is equal to the angle, i.e. $\frac \pi 3$, and there are six of these lengths in the circumference $2\pi$ of the circle.)