Why are turns not used as the default angle measure? Why is $2\pi$ radians not replaced by $1$ turn in formulas?
The majority of them would be simpler. If such a replacement was proposed earlier, why was it declined?
 A: In fact, "radians" are not a unit like meters or second, so you can't rescale them to make $2\pi$ become $1$ (as sometimes you do in physics, rescaling e.g. meters to make $c=1$).
As an aside, degrees are an absolutely artificial concept, and you should try to never use them.
A: Because if you use $1$ for the turn instead of $2\pi$ (forgetting what a radian is), you get that $\frac{\mathrm d}{\mathrm dx} \sin x = 2\pi\cos x$, which is probably not what you want, and many other problems, such as
$\sin x$ not being a solution to $y''+y=0$ etc.
You can discuss whether the full turn is $2\pi$ or $\tau$ (tau), but you can't change the fact that it's equal to $6.2831853071\cdots$.
A: Radians are the natural, dimensionless choice of unit to measure angle. We could certainly define the full turn as $1$. That's a nice, wholesome unit. It's nicer than pushing our historical comfort with base $60$ with $360^\circ$ or dealing with cumbersome $400 \text{ gon}$. But does $1$ really reflect the system we're trying to describe? 
In choosing an appropriate unit, we should ask "What is an angle measure?" One appropriate definition is the measure $\theta$ of the angle subtended by an arc length $s$ of a circle of radius $r$. This definition, though, needs some work to become a practical tool. Given the obvious relationship between an angle and a circle, we can call upon Euclidean geometry to find ourselves a nice parameter. Behold the ratio of circumference to diameter: $C/d=\pi$! Now we observe that the relationship between arc length and circumference goes as
$$s = (\text{some fraction})\times C = \dfrac{\theta \times C}{\theta_{turn}}$$
We're getting somewhere. Let's adopt a convenient parametrization of angle measure, something we can throw numbers at and interpret easily. Perhaps a convenient choice would necessitate a convenient circle. Well, the unit circle has a nice radius and area. Its circumference is $2\pi$. If the distance around the unit circle is $2\pi$, then why don't I adopt this for my unit of angle measure? I'll define the angle measure $\theta_{turn}$ that takes me around the circle to be $2\pi$. 
Arc length is now cleanly given by
$$s = r\theta $$
That's the definition of radian in that last line. Why is it a superior choice? Any other choice of units would have left over some other constants. With the choice of $2\pi$, we have a clean, relevant, and dimensionless unit.
A: Just a complement to zahbaz answer.
With angles in radian, you get the nice formulas:
sin' = cos
cos' = -sin

sin(x) = 1/2 * (exp(ix) - exp(-ix))
cos(x) = 1/2 * (exp(ix) + exp(-ix))

Of course those are mathematical considerations, and that's the reason why in common life we use degrees.
But there are other units. In artillery, the thousandth (millième in french) is used. You have 6400 of then in one turn (why not...) but as an approximation, it gives an elevation of 1m at 1km of distance. If you have this in binoculars and can guess the height of something, you can easily compute a distance - even a military can...
A: There is a movement to make $2\,\pi=\tau$ a fundamental constant instead of $\pi$ (read The Tau Manifesto.) But not as far as I know to use $1$ as the measure of the whole circunference. One possible reason is that $\pi$ encodes the relation between the radius an the length of a circumference: a circular sector of radius $r$ and angle $\alpha$ radians has a length equal to $\alpha\,r$; it would be $2\,\beta\,\pi\,r$, where $\beta$ is the measure of the angle in the units you propose. Moreover, you cannot avoid $\pi$ in formulas like
$$
e^{\pi i}+1=0.
$$
