Although B is entitled to start a move when A has released a legal move, he was in fact not entitled to move because A's move was illegal. Therefore B's move has not been "made" and B is still entitled to claim the illegal move of A.
So, let's give a scenario and see what happens:
1. Player A has the move. Player A shifts a piece illegally, say a bishop, but doesn't press the clock, then
2. Player B picks up a piece, say a rook, with the intention of moving.
Isn't Player B breaking the laws. Because, as you have both agreed, Player A has not “made a move” and therefore Player B “does not have the move”. Shouldn't Player B's action in picking up his rook be regarded as distracting behaviour as Player A is still in the process of making a move.
3. Now, quickly, Player A shifts the bishop to a square checking Player B's king, making a legal move. Player B still has his rook in his hand.
Isn't Player B required now to make a move with the rook, if it is legal to do so?