There's good reason to define the radian as arc length/radius rather than sector area/radius squared, even if you don't count the fact that arclength is 1 dimensional and thus the most fundamental measurement or the factor of 1/2 in sector area coming from triangles.
Many things in trigonometry only work with radians and not any other system of measurement. With radians, cosine is the derivative of sine.
The Taylor series for all the trig functions and Euler's formula only work in radians. These things wouldn't work with areans. So changing radians won't work. If could just use any unit we wanted, we would probably just use revolutions, but unfortunately, there are certain things that radians are needed instead of rotations. (Luckily, though, we have a convenient conversion factor between revolutions and radians called tau).
Also, even if we redefined radians, Euler's identity wouldn't change because complex exponents still have the same values; all that would happen is that Euler's formula would become much more messy. We know Euler's formula works in terms of radians because by substituting iθ for x in the Taylor series for e^x, you end up with the Taylor series for cosine plus i times the Taylor series for sine, but those Taylor series only work in terms of radians, so Euler's formula only works in radians. Therefore e^(iπ) and e^(iτ) still have the same values. However, if you do want to have the perfect e^(iτ)=1 instead of e^(iπ)=-1, just use tau instead of pi.
So what is the real source of the conflict? The real problem is a bad definition of pi. Pi's definition of circumference divided by DIAMETER is inconsistent with the definition of radians, arc length divided by RADIUS. Tau's definition as circumference divided by RADIUS is consistent, however.
Also, when you said radians are not defined like the arc of a circle, that's wrong. Radians are defined as arc length divided by radius. A one radian sector has the extremely elegant definition as a sector whose boundaries are all the same length, and these boundaries are two radii and an arc. What you want to change the definition to is to have them defined like a sector, not like an arc.