Why is an angle defined as the ratio between the arc length and the radius, and why is arc length defined as the product of the radius and an angle?

I'm having a hard time understanding why

$$\theta = \frac{s}{r}$$

and

$$s = \theta r$$

where $$s$$ is the arc length, $$r$$ is the radius, and $$\theta$$ is the angle measure.

I understand how one is derived from the other through algebraic manipulation.

I'm trying to understand intuitively:

1) Why dividing some amount of the circle (arc length) by the length of its radius would dictate how much an angle $$\theta$$ "opens up." Since it's a ratio of two lengths, then why would the result be an angle measure?

2) Why multiplying the radius by an angle measure would give the arc length. I'm having a hard time grasping this one because the angle measure and the radius seem like (for lack of a better term) two different "units" to me. An angle $$\theta$$ is a measure of angular rotation whereas the radius is a length, so how does multiplying the two then give back another length i.e. the arc length?

As is well known $$\pi$$ is defined as the ratio between the perimeter of a circle and its diameter or 2 times the radius which is the same, so $$\pi = \frac{p}{2r}$$. So $$p = 2\pi r$$. So as $$p$$ is the whole circunferences then the half must be $$\pi r$$, $$\frac{2}{3}$$ of circumference must be $$\frac{4}{3} \pi r$$, and so on for every fraction of the perimeter. Now we defined the angle as the length divided by the radius because of this fact, because what $$r$$ does is just scaling the part of the circumference, but the angle remains the same.