Using monetary prizes bigger than the value of all the wealth of the planet is the cause of the confusion.

I don't think it is. The cause of the confusion is just that the sums are wrong (and the conclusion is wrong). Replace the opening statement with "I'm Bill Gates, I'm offering you a bet - the cost to take the bet is $1, the prize for winning is $58 billion. The odds of winning are 1 in 57.99999 billion".

Now we're no longer talking about unrealistic amounts of money, but it still isn't good bet for Bill to offer, because it's expected value is negative. You do need to invoke the fact that wealth is finite to explain why martingales) don't work, but this "system" isn't nearly as complicated as a martingale.