mathematics - Why are radians more natural than any other angle unit?

I'm convinced that radians are, at the very least, the most convenient unit for angles in mathematics and physics. In addition to this I suspect that they are the most fundamentally natural unit for angles. What I want to know is why this is so (or why not).

I understand that using radians is useful in calculus involving trigonometric functions because there are no messy factors like $\pi/180$. I also understand that this is because $\sin(x) / x \rightarrow 1$ as $x \rightarrow 0$ when $x$ is in radians. But why does this mean radians are fundamentally more natural? What is mathematically wrong with these messy factors?

So maybe it's nice and clean to pick a unit which makes $\frac{d}{dx} \sin x = \cos x$. But why not choose to swap it around, by putting the 'nice and clean' bit at the unit of angle measurement itself? Why not define 1 Angle as a full turn, then measure angles as a fraction of this full turn (in a similar way to measuring velocities as a fraction of the speed of light $c = 1$). Sure, you would have messy factors of $2 \pi$ in calculus but what's wrong with this mathematically?

I think part of what I'm looking for is an explanation why the radius is the most important part of a circle. Could you not define another angle unit in a similar way to the radian, but with using the diameter instead of the radius?

Also, if radians are the fundamentally natural unit, does this mean that not only $\pi \,\textrm{rad} = 180 ^\circ$, but also $\pi = 180 ^\circ$, that is $1\,\textrm{rad}=1$?

Answer

Angles are defined as the ratio of arc-length to radius multiplied by some constant $k$ which equals one in the case of radians, $360/2\pi$ for degrees. What you're effectively asking is what's natural about setting $k$ = 1? Again it's tidyness as pointed out in dmckee's alternative answer.