How and why were the chess pieces given as "odds" predetermined? For instance, "pawn odds" is really "pawn and move" odds. That is, the odds giver plays Black and removes his KBP.

Why not allow an "intermediate" odds where the odds giver plays white? And why not allow the odds give to choose which pawn to remove?

In the case of knight and rook odds, it is the queen knight or rook that's removed, but why not allow the odds giver to substitute the king's pieces instead?
I would like to challenge the premises in the question:


  Why not allow an "intermediate" odds where the odds giver plays white? ...In the case of knight and rook odds, it is the queen knight or rook that's removed...


Emmanuel Neiman cites the following example in Tune Your Chess Tactics Antenna:

[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKB1R w KQkq - 0 1"]
[White "Kennedy"]
[Black "NN"]

  1. e4             e5            
  2. Bc4            Nf6           
  3. d4             Nxe4          
  4. dxe5           Nxf2          
  5. O-O            Nxd1          
  6. Bxf7+          Ke7           
  7. Bg5#            1-0


Admittedly this game is played in the nineteenth century, but it is a counterexample to the premises in the question.

Seeing as handicaps are not in the FIDE rules, then it is for the players to agree as they see fit, and in this example the players agreed that white would be without his kingside knight.