The average of the 3 points would give you the triangle's centroid, not necessarily its circumcenter. You can still calculate the circumradius from a, b, and c: https://mathworld.wolfram.com/Circumradius.html.
The point of the article is to point out the non-obviousness of some true statement that an outsider to mathematics would take to be obviously true, and thus to demonstrate the necessity of formal (or at least careful) reasoning in mathematics. It's posed as the question of whether the square root of 2 exists, but really, it should be posed as the question of whether there exists a unique real number whose square is 2. Of course, if the square root of 2 didn't exist, we could just define it as an object whose square is 2 and be satisfied like we might for the square root of -1. However, if we did that, our new sqrt(2) wouldn't be a real number.
To answer the better-specified question we need first to define the real numbers and what it means to multiply two real numbers together. This would be done in a real analysis class, after which one could properly prove that sqrt(2) exists, and understand what the proof actually accomplishes.
I believe that the article poses the problem as whether sqrt(2) exists and not as whether sqrt(2) is a unique real number because the article is meant to appeal to the outsider who wouldn't see the point of the quantification, at least, not before reading the article.