360 is an incredibly abundant number, which means that there are many factors. So it makes it easy to divide the circle into $2, 3, 4, 5, 6, 8, 9, 10, 12,\ldots$ parts. By contrast, 400 gradians cannot even be divided into 3 equal whole-number parts. While this may not necessarily be why 360 was chosen in the first place, it could be one of the reasons we've stuck with the convention.
By the way, when working in radians, we just "live with" the fact that most common angles are fractions involving $\pi$. There's a small group of people who prefer to use a constant called $\tau$, which is just $2\pi$. Then angles seem naturally to be divisions of the circle: The angle that divides a circle into $n$ equal parts is $\tau/n$ (radians).
Hope this helps!