If people hold onto money, then they are not spending it on consumer goods or saving it in which case the money gets spent on producer goods. The reduced demand for goods lowers their prices. Lower prices for goods means a larger purchasing power of money. The larger purchasing power of money (i.e., the higher price of money) is what satisfies people’s desire to hold more purchasing power in the form of money even though the stock of money has not changed.
If we were to diagram the analysis, the increase demand for money shifts the demand for money curve upward to the right, which results in a higher price for money (in the link below ignore the Keynesian assumption that the price of money is the interest rate). The higher price for money eliminates the excess stock of money which the increase demand for money (the difference between a and b) would bring about in the absence of a higher price for money.
On average people are worse off over the cycle because there are fewer goods produced in the economy than if there had been no cycle. Of course, some workers might be better off and others worse off depending on the relative movement of demand for their labor and some entrepreneurs may be better off and others worse off depending on the relative movement of demand for their goods. But overall, society is worse off.
As you say, people are not worse off or better off because prices and wages both have fallen.