I know the Georgism joke here, but we don't need an LVT to fix this; we just need states to override home rule on minimum density requirements for residential and mixed-use zoning. The core problem here is in fact very simple!

A land value tax without changes in non-tax land use regulations would not fix, and a land value tax would not be necessary after removing the non-tax land use restrictions which this responds to.

Not on its own. You also need to relax or remove direct (and indirect, eg parking minimums) restrictions on residential density.

How?