You forgot the square root on distance formula. Distance = square root of ((X1 - X2) ^ 2 + (Y1 - Y2) ^2 + …).
Consider the origin (0,0,0, …) to a random point on the sphere (~0, ~0, ~0, …). So Distance from origin = square root of ((~0-0)^2 + (~0-0)^2 + (~0-0)^2 + … ), which sums to 1 by definition of the unit high dimensional sphere.
Then plug in 1 vs 0 in the first place because we care about (1,0,0,0 …) and you get the correct answer = square root of ((~0-1)^2 + (~0-0)^2 + (~0-0)^2 + … ) ~= square root of 2.
Consider the origin (0,0,0, …) to a random point on the sphere (~0, ~0, ~0, …). So Distance from origin = square root of ((~0-0)^2 + (~0-0)^2 + (~0-0)^2 + … ), which sums to 1 by definition of the unit high dimensional sphere.
Then plug in 1 vs 0 in the first place because we care about (1,0,0,0 …) and you get the correct answer = square root of ((~0-1)^2 + (~0-0)^2 + (~0-0)^2 + … ) ~= square root of 2.
Edited to fix typo and add clarity.