What makes radians superior to turns/revolutions?
The actual definition of the complex exponential map is equivalently via an ODE ($f'=f$, $f(0)=1$) or via power series ($f(z)= \sum_{k\geq 0}\frac{z^k}{k!}$).
Similarly, cosine and sine functions are defined via power series above all. If you are to change the way you measure angles and therefore define a new, distorted version of sine and cosine, then the most natural thing to say would be that the Euler formula simply doesn't hold in your new settings. Or you could use it as a new definition for a distorted exponential map, but then what is more important for you: the fact that Euler holds, or the fact that $\exp$ satisfies the above ODE?
All of mathematics rely on the fact that we agree on axioms, language and definitions. How natural these are depend on how beautiful the resulting theorems seem to the trained eye.
The main issue here is that Euler's formula is correct for radians only. Your questions are arising from using it with turns instead of radians, but you're starting from a false premise.
Often, the order we learn things isn't equivalent to the most precise mathematical definition of things.
For example, at school we first learn exponentiation with natural exponents, then we generalize for integers and even rationals, but how about irrational exponents, for example? The best path to follow if you're looking for precise math (and that's what I suggest you do here) in this case is to consider the definition of the exponential function as the nonzero function that satisfies the property $f(x+y) = f(x)f(y)$. And then if you go further to complex numbers, you'll see that actually one needs to choose what is called a branch of the logarithm function, and so on.
In fact, $2^i$ can have more than one value, depending on the branch choice, but usually we take the so-called principal branch. Regardless, this is all a set-up to reiterate that if you go to the very definition of complex exponentiation / principal complex logarithm, you'll see that Euler's formula only applies to radians in the first place, so your premise is false.
While Arnaud Mortier provided very good answer, I'd like to make another very simple observation - that you consider blindly changing of the units of only some of the values, but forgot to check&update the formula/constants/etc.
This is a thing you always need to keep track of when dealing with physics, electronics, etc, where it's common to use various units for the same kind of values, and often we keep unit markings all the time (think Ohm: not just 10/0.5=10, but 5V/0.5A => 10Ohm, etc) especially to not forget and mix them up.
I see that you've tried to formalize the question, so forgive me if my answer is too naive or too lax..
But, let's see. For generic typical (planar/Euclid/..) triangle:
$${\alpha}+{\beta}+{\gamma} = {\pi}$$
If I use radians for all input angles, all's fine:
$$\pi/4+\pi+\pi/4 = {\pi}$$
But If I try to use turns or degrees, everything detoriates:
$$45+90+45 = {\pi}$$
That's unless the PI constant adapted to turns/degrees as well.
$$45+90+45 = 180$$
But that's the easy case. Sum is linear operation, so i.e. changing units ("scale", "multiplier", etc) on one side implies need for just changing units on the other side (a+b=c => (kx+ky)=c => x+y=c/k => x+y=z // a=kx,b=ky,c=kz, all units kept same, you may do it "blindly")
In your case, you started to wonder about using turns and degrees in:
$$e^{i\theta}=\cos\theta+i\sin\theta$$
but:
- first of all, there are some constants (e, i) and you have not even tried to update them to new units
- more importantly, some of the operations are not linear (exponent), hence units will not "just transfer to the other side"
This just reminds us to keep the formula up-to-date with our assumptions. Now, if we keep the units in mind, we might update the formula:
$$e^{i(\theta/2\pi)}=\cos(\theta/2\pi)+i\sin(\theta/2\pi)$$
which would work on turns.. I suppose that's pretty obvious and simplistic, but that's because I chose the simplest possible adaptation, just because I wanted to show the point why units matter.
Another thing I just noticed, you've said about the sine/cosine:
sin(ln2) will be the y coordinate of the unit circle when the angle is 0.693 radians. But if we use turns instead of radians, then sin(ln2) will be the y coordinate of the unit circle when the angle is 0.693 turns
IMHO, again, dropping units forced you into error.
If we define sin/cos in terms of abstract numerics, we often conventionally use radians. However, at least in my country, in primary school, concept of "pi" and "radian" is a bit too much, so they use degrees (0°-360°) and sin/cos are defined in terms of degrees (a.k.a "deg scale"), so sin(0) = sin(180) = sin(360) = 0, sin(90)=1, etc.
Note how that's ambigous to other readers? If I didn't state it explicitly, the 360 could be as well read as radians? It's easy to tell that 360 is probably degrees, but how about 1.123?
If we now truly keep units in mind, I should have written it unambiguosly, something like: sin(0°) = sin(180°) = sin(360°) = 0, sin(90°)=1, etc - if I assumed sin/cos to be defined in degs.
On the other hand, if I assumed sin/cos to be defined in rads, then I would have to translate all constants accordingly (0,π,2π,π/2), or "update the formula" and write sin(0°/(360°/2π)) = sin(180°/(360°/2π)) = sin(360°/(360°/2π)) = 0, because now passing a degree, or turn, would be out of its defined domain..