>Therefore, to approximate the value of √2, we can simply measure the time it takes for a player to travel at a constant rate along one of the legs and the time it takes for that player to travel along the hypotenuse.

In many games, moving diagonally (e.g. by pressing Left and Forward at the same time) gives the player a higher speed than moving orthogonally. The sideways and forward movements are added, rather than normalized, and so you could traverse the hypotenuse in the same amount of time as traversing a leg. I know this was the case at one time in Minecraft, but not sure if it still is. Presumably they did their measurement by just facing along the hypotenuse and walking forward, but it's the sort of weird game mechanic you have to be on guard for when doing this sort of thing.