What is the significance of deciding the convention of $1 \text{ radian} = 180 \text{ degrees}$ over $\pi$?

From a mathematical point of view you are asking the question the wrong way around. You seem to be assuming the degree is fundamental and the radian is derived from it to be simpler and get $\frac {180}\pi$s out of the equations. From a mathematical view the radian is the fundamental unit and the degree is something in the range from a derived unit to a bad mistake. From an engineering and conceptual point of view degrees are nice because they give simple measures to the parts of a circle we use most (except for my dad who had to cut round desserts into seven pieces)

Historically speaking, it is not likely that beautifying the Taylor expansion of sine was “the” reason for defining the radian. After all, the radian was defined by the relationship $r\theta=s$, which is incredibly helpful in physics. Perhaps un-coincidentally, the man who defined the radian in the 1870s, James Thomson, was the brother of the famous physicist Lord Kelvin.

The English mathematician Brook Taylor died almost 150 years beforehand, so I suppose that Taylor expansions could have been considered. In fact, this is one of the reasons we continue to use the radian today. Only when the arguments of sine and cosine are expressed in radians is it true that $\frac{d}{dx}\sin x=\cos x$ and that $\frac{d}{dx}\cos x=-\sin x$, and these relationships are how the Taylor expansion of sine is defined.

The big restriction is the differentiation formulas for $\cos(x)$ and $\sin(x)$ only hold when $x$ is measured in radians. The definition of the radian is as follows : 1 radian is defined to be the angle that subtends an arc of length 1 on the unit circle. Using similarity of circles, you get the equation $$s = r \theta$$ where s is the arc-length and $\theta$ is in radians. An arc length of $2 \pi r$ corresponds to the entire circumference of a circle. Plugging in gives$$ 2 \pi r = r \theta \implies 2 \pi = \theta$$ Thus there are $2 \pi $ radians in a circle. Equating $2\pi \text{ rad }= 360^\circ$ gives you the conversion factor between radians and degrees.

So the definition of the radian had absolutely nothing to do with degrees. Defining the radian in the above way gives you the slick formula for derivatives of sinusoids.

Each definition was independent and it turned out that $1$ radian happened to be the same angle as $\dfrac{360^\circ}{2\pi}= \dfrac{180^\circ}{\pi}.$

In one instance you are dividing the circumference of a circle into $360$ evenly-lengthed segments and in the other you divide it into $2 \pi $ segments.