However, the market could normalize after this as the money earnt by a nation's richest would eventually be more evenly distributed amongst others, hence they would spend more, canceling the downturn.
Wage cap is not an redistribution measure. It’s an measure aimed at reducing inequality but without any direct redistribution.
For example, suppose you have Peter and Paul, Peter earns 10000\$ per month and Paul 1000\$ per month. If there is wage cap 5000\$, Peter will earn 5000\$ and Paul is still earning only 1000\$. Hence there is no direct redistribution effect even though inequality is now lower.
So there will not necessarily be any more spending in an economy. That assumption is completely false and would hold for income tax that redistribute money from rich to poor but not for wage cap. In fact most likely there will be less spending in the economy as result of wage cap.
Regarding other long term effects of wage cap, it would severely depress the labor supply of people to whom it applies to. Wage cap is de facto equivalent of confiscatory tax that confiscates any income above certain level, but it is even worse as government does not even get any revenue, from it. If a person does not earn any income for their labor above certain level they will have no incentive to work anymore (aside from maybe the fact that they enjoy their work). Hence their labor supply will be depressed. They will spend less time at work and more time enjoying some leisure activities.
This can have serious impact on economy as usually people who earn more in market based economies are those people who are more productive. Highly skilled doctors, managers, scientists etc, decreasing overall productivity of the economy. Moreover, such measure would create a permanent supply shock (as long as it would be in effect) permanently lowering the level of GDP and hence also the material welfare of people in given country.
It could also impact the growth of the economy, as endogenous theory of economic growth says that human capital formation can be very important driver of economic growth and since people tend to earn more the more human capital they have this would depress the human capital formation for the very brightest who would be affect by such a measure.
Also as a side note, I don’t really know any public economists who would approve of wage cap. The reason for that is that in fact if you care about redistribution you want as many people with high incomes as possible so then you can tax their incomes and raise the welfare of the poor through redistribution.