There is an argument for the view that a constitutional monarchy, serving as a living symbol of the legal continuity of the state, can ease the acceptance of social changes - by absorbing them into the national tradition it symbolises. Politicians do the leading from the front in debates (theoretically), royalty’s job is to stay near, or even slightly behind, the centre of gravity of public opinion. Such causes as they are associated with are what I think the US would regard as “First Lady” causes.
Taking Denmark specifically, the last time a monarch tried to interfere politically (in 1920) he was rebuffed. Famously, Juan Carlos of Spain intervened to stop a military coup d’état against the then new democratic régime in 1981, but that’s a rare example in exceptional circumstances.
As for private wealth, high levels of tax and public expenditure don’t necessarily preclude private wealth - they might restrain its acquisition and inheritance, but they’re not deliberately confiscatory.