Indeed, there is no distribution in which the probabilities are both 1/2. Such a distribution does not exist so the probabilities are never 1/2. So the paradox does not arise at all.

If you change the game slightly to the following: You receive X amount of money and you can flip a coin. If you flip Tails they have to pay you 2X, if you flip Heads they have to pay you X/2, obviously you always should flip because in that game, the probabilities are 1/2.

Indeed, there is no distribution in which the probabilities are both 1/2. Such a distribution does not exist so the probabilities are never 1/2. So the paradox does not arise at all.

If you change the game slightly to the following: You receive X amount of money and you can flip a coin. If you flip Tails they have to pay you 2X, if you flip Heads they have to pay you X/2, obviously you always should flip because in that game, the probabilities are 1/2.