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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?
The queenside pieces are less missed than their kingside equivalents, not only in defense but also in attack. Giving the kingside piece amounts to increasing the handicap.
This would be most pronounced if the bishop's pawns were involved, because the square on which they originally stand is the weakest point in the uncastled position, and vacating it offers open lines for an attacker to check the king both before and after castling, and may result in being unable to castle at all.
So, while it was done to give the kingside piece, the queenside piece was considered fair odds.
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.