Limiting free parking works as part of a long-term solution to make driving a car in high-density areas miserable and expensive, while making walking, cycling or taking public transportation enjoyable and cheap.

Other than making parking expensive, I'm these include: closing roads; creating roads closed to private vehicles, allowing only electric busses/trams; city bike programs; increase in pedestrian and cycling infrastructure; closing side roads on one end to remove their functionality as transit roads; congestion charges; limiting access to the center to odd numbered plates; requiring residential buildings to include ground-level commercial spaces; narrowing roads; increasing the number of buses/trams; park and ride systems; car shares to help people living in high density areas to abandon car ownership; etc.

You can see every single one of these solutions implemented in cities all over Europe. Europe is a obviously a different case than America - almost all of its cities had centers whose roads were defined before cars were invented. After 60+ years of trying to shoehorn car culture into them, they've said "enough". They collectively decided that they literally can't afford to accommodate cars at a rate that drivers want, and decided to take another route. Which is actually an interesting phenomenon once you start noticing it - there is no infrastructure that you can build in a high density area that will not be completely filled with cars within a couple of years. The only place where high car ownership makes sense is in rural areas.