Of course, my calculation of mobility that takes into account the average fullness of the board is an important step in this direction. Such a calculation would be an important part of estimating endgame values, but, interestingly enough, it doesn't tell the whole story.
A large part of the Rook's special endgame value comes from the way that its power of movement cooperates with the move (but not the capture) of the Pawn. The Rook can sit behind a Pawn and "push" it forwards.
In addition to that, the Rook has the special power of blocking the enemy King, by controlling a rank or file that cuts the King off from the rest of the board.
In contrast to the Rook, a KnightRider (N7 in my funny notation) has less empty-board mobility than the Rook, no special ability to co-operate with Pawns, and no ability to interdict the enemy King.
If you look at a random endgame where each player has an equal number of Pawns, from 4 to 6 Pawns for example, and the Pawns are spread around on both sides of the board, and one side has a Rook, the other side has a KnightRider, and there are no pieces on the board other than the Kings, what do you find?
The KnightRider has an advantage, usually a winning advantage.
It's purely a matter of technique, and if you're not familiar with the technique you may think that the Rook has the advantage. Simply put, the KnightRider has so much forking power that sooner or later it will be able to give check while attacking an undefended Pawn (or the Rook); and in order to make this happen, you move the KnightRider around constantly to different squares attacking different things and forcing the enemy into different defensive positions, until at last something falls.
(Actually, this is quite a beautiful endgame, and I was very pleased when I found the proper technique.)
When you get down to a smaller number of pieces, say Rook and Pawn versus N7 and Pawn, it seems that the Rook has the advantage. It's enough to make you wonder how "end"y an endgame needs to be before endgame advantage becomes apparent!
You could explain the strange case of the KnightRider as a demonstration of the value of moving in 8 directions rather than in 4 directions.
However, part of the value of a piece depends on the composition of the rest of the army, the role of the piece in the army, and the piece's starting position on the board.