I agree that this is not within the reach of most high school students, and if I had to guess, I think that either these Moscow kids were very carefully selected to be in this group or most of them didn't follow it.

But I do think some very talented and motivated high school students could at least follow the gist of this argument! I've worked at a summer program for mathematically talented high schoolers, and while I didn't teach this particular class, one of the years I was there someone taught what I think is essentially the same proof this article is talking about, and they did get through it in five weeks.