Has this simple proof appeared in literature? This is essentially the same proof as one where you call the vertices $\vec{a},\vec{b},\vec{c}$, and observe that $\frac{\vec{a}+\vec{b}+\vec{c}}{3}$ lies on each of the three medians.
Consider the triangle $\bigtriangleup ABC$ with the sides $AB,BC,CA$ having midpoints $D,E,F$ respectively.
Imagine that the triangle is split up into infinitesimally wide strips, each of whose long sides are parallel to the side $AB$. The center of mass of each of these strips lie on the median $CD$. Hence, the center of mass of the entire triangle also lies somewhere on the line $CD$.
By an identical argument, the center of mass of the triangle lies somewhere on the median $AE$ as well as somewhere on the median $BF$.
Thus in fact, all three medians $AE,BF,CD$ have a common point, which is the center of mass of the triangle!
Googling triangle median centroid infinitesimal yields this excerpt from page 69 in the 1917 edition of Applied Mechanics by Alfred P. Poorman (highlighting by Google Books):
The proof doesn't explicitly assert that the medians are concurrent, though it's implied by "Therefore the centroid is on any median". (Applied Mechanics is a free ebook!)