Be careful with your last paragraph. It builds upon results about the relationship between trigonometric functions (usually defined in terms of power series) and the unit circle's geometry, as well as intuition from 2D geometry in general. These things can actually be quite hard to prove. (Prime example in a slightly different context: https://en.wikipedia.org/wiki/Jordan_curve_theorem ) For instance,
> A similar triangle inscribed at this vertex has a short side which is necessarily less than the arc length.
already uses the fact that straight lines are the shortest lines in the class of C¹-differentiable curves due to the Euler-Langrange equations. (Good luck with proving this statement in the class of C⁰ curves.) Then, to use this fact, you need to prove differentiability of a curve parametrizing the unit circle, i.e. differentiability of the trigonometric functions. Now, at this point you might as well use the standard definition of e^x in terms of a power series because it simplifies the differentiability proof tremendously.
You can get that the interior line is shorter and the outer line is longer than the arc by relating these to the inner and outer polygons used in Archimedes' demonstration that the area of a circle may be related to the circumference:
It is possible to prove that the circumference of the outer polygons gives a decreasing sequence and the circumference of the interior polygons gives an increasing sequence simply by applying the triangle inequality when you double the number of sides. This is sufficient to justify the use of the squeeze theorem here, although you are correct that the axiomatization of geometry is quite subtle.
I did not give much thought to the definition of the trigonometric functions while writing this because I assumed this theory is usually fully developed before we begin complex analysis (but is it?).
> A similar triangle inscribed at this vertex has a short side which is necessarily less than the arc length.
already uses the fact that straight lines are the shortest lines in the class of C¹-differentiable curves due to the Euler-Langrange equations. (Good luck with proving this statement in the class of C⁰ curves.) Then, to use this fact, you need to prove differentiability of a curve parametrizing the unit circle, i.e. differentiability of the trigonometric functions. Now, at this point you might as well use the standard definition of e^x in terms of a power series because it simplifies the differentiability proof tremendously.