You are 100% correct that if a position is ever reached where everything has
returned to the initial position, white must have the move.
But note that I asked if white has the move any time a position occurs where everything is in its initial position, and so we must also consider the initial position before any move has made.
When the pieces are in their initial positions before the first move has made, does white have the move? You would think so, but actually that is not the case. Nor does black have the move.
The FIDE Laws of Chess define having the move thusly:
> The game of chess is played between two opponents who move their pieces alternately on a square board called a ‘chessboard’. The player with the white pieces commences the game. A player is said to ‘have the move’, when his opponent’s move has been ‘made’.
Nobody has the move until while makes the first move, then black has the move.
This stupid way FIDE defines having the move can show up in another situation. Consider this game:
1. Nf3 Nf6
2. Ng1 Ng8
3. Nf3 Nf6
4. Ng1 Ng8
The initial position has now occurred 3 times (at the start, after move 2, and after move 4). Can white claim a draw by threefold repetition? Could black have claimed a draw before making 4. ...Ng8 by writing it on their scoresheet and informing the arbiter?
With the way FIDE defines having the move, the answer should be no. This is not yet threefold repetition. Part of FIDE's definition of a repeated position is that the same player has the move in both, and so the positions after move 2 and move 4 are not repetitions of the position at the start because white has the move in the later two but no one had the move in the former.