"When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right" is one of the opening definitions of Euclid's Elements. Why did Euclid, or rather Pythagoreans he was systematizing, single it out? Well, it was not by averaging.
For one, the right angle is easy to define, with only two straight lines and equality as prerequisites. For comparison, one would need three straight lines to define $60^\circ$, and more for other alternatives. Dropping perpendiculars, that come out of this definition, is also a ubiquitous step in Euclidean demonstrations.
For two, as one can see from the opening propositions of the Elements, Pythagoreans were interested in angles, first and foremost, for their use in triangles. And the right angle produces the simplest form of the relation among the sides of a triangle having it, $a^2+b^2=c^2$, the crown jewel of Pythagorean geometry. For comparison, if the angle were $60^\circ$, say, it would have been
One could, perhaps, instead favor as the primitive what is sometimes called "straight angle" ($180^\circ$), definable with a single straight line. Today. But back then it was quite an unnatural "angle" to consider, as it happened before the advent of abstract mathematical formality that embraces extreme cases and extensions for the sake of generality. Indeed, Euclid and other ancients never talk of "straight angles", let alone greater ones, just as they never talk of zero and negative integers. Those creations came later. When Euclid needs to, as in the parallel postulate, he says "two right angles":
"If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles."
Also, bisection, needed to go from $180^\circ$ to $90^\circ$, is more complex than duplication, needed to go the other way. So if one intends to use both $90^\circ$ may still be easier to start with.
Regardless of whether one finds such arguments from simplicity compelling, by the time of the French revolution the force of Euclidean tradition was by itself more than enough to make the right angle "quintessential" and a natural choice for a primitive.