What would happen? The same thing that has happened everywhere else LVT has been tried: it quickly becomes watered down until the tax treatment of most property owners (especially homeowners) effectively returns to the status quo ante.
LVT isn't politically viable, particularly in contexts (such as the U.S.) where property taxes are principally used by state and especially local government for revenue.
Note that Prop 13 largely passed because 1) people were concerned about seniors being forced out of their homes and 2) the informal practice of property assessors undervaluing the homes of seniors resulted in a perception of unfair and capricious assessments. LVT is a radical move in the completely opposite direction.
The Obamacare debate has shown how irrationally attached people are to specific doctors, and how reactionary they get when you expose them to market forces in an attempt to improve overall welfare. But that's nothing compared to the reaction that happens when tax policy systematically and (more importantly) visibly forces people to move out of their homes. Even if LVT was in practice Pareto Optimal--all people forced to move were objectively better off (e.g. significantly richer)--people would never perceive it that way.
If something isn't politically viable, theoretical arguments are pointless. That we've discounted, forgotten, and even rejected that basic truism of governance--politics is the art of the possible, the attainable--explains much about today's divisive and ineffective politics.
A corollary is that you can't separate politics from economics. Political power is a form of currency--votes in a democracy being the most obvious medium--used for exchange just like money. It's just that what people are exchanging (favors, policies, dreams) is much less concrete than physical goods or interpersonal services.
More than most economic policies LVT utterly fails to account for this aspect of market behavior. LVT only efficiently internalizes costs and benefits if we ignore politics. But it should be obvious that the character of taxation and process of assessment would result in people very quickly diluting LVT. Taxation and property assessment are entirely government functions, LVT invites voters to effectively rent seek en masse, LVT provides no market mechanism whatsoever to compensate, so an LVT regime is unsustainable where you have a free political market.