Density.

The current property tax system makes it more likely that (in your example) the law firm will push out the bakery. Because improvements are taxed adding more commercial space means paying more taxes, which makes it less likely to happen.

With an LVT, expanding the building (up or out) to add a second store/office) is cheaper so you’re more incentivized to provide space for both

Income per square foot in _rent_.

The important thing isn't that the lawyer's office makes more money. That's just correlated with the important thing, which is that the lawyer's office is willing to pay more rent. Note the lawyer isn't operating a charity for property owners here; they're only going to pay as much for rent as required for the market.

If prevailing rent is high enough such that only lawyers get retail square footage, then clearly property should be taxed as such. That's the value of the land. But then more development will happen, eventually reaching an equilibrium.

Saying rent isn't quite right. These land value taxes are usually taxed on potential as zoned. Empty land isn't bringing in rent. The whole point of these land value tax schemes is to tax it based on its highest zoned value so you force (re)development of the highest use case.

> Let's say because of rising taxes, a bakery or a shop has closed and a lawyers office has opened in that spot.

Taxes will rise in neighborhoods where improvements have happened. To determine the neighborhood is improving, we need a metric, for instance traffic. We expect the donut shop in an improved neighborhood to see increased traffic and more sales. If traffic doesn't increase then land value hasn't actually increased and the taxes have been applied inappropriately. Maybe we can come up with a better indicating mechanism for land value, I admit I have not read Henry George's work.

In your example, bakeries usually don't own land in highly desirable cities, they lease it from the owner. I think that's fine. The incentives here are against huge capital behemoths that can buy land and afford not to develop it, costing the cities in potential consumer revenue, population growth, broken windows theory, etc. Lawyers might own the building and be able to afford the tax. I think it's fine to expect bakers cannot afford buildings in very expensive cities (unless you're some luxury brand). If owners are incentivized to develop and lease their properties, there should be more supply for a cheaper rental market.