If you make a revenue-neutral shift in the tax share from income to real estate, real estate prices are going to drop.

Not so fast.
First, of course tax increase is a market value depreciation. But only half of the housing is apartments, another half is commercial real estate. And from both these big portion are not owner-occupied. And add there natural monopolies.
But second, real estate tax transfers money from economic rent payments to the demand of goods and services. This means higher employment, higher wages (=less labour supply) and higher wealth for wealth creators. This would change income distribution for the benefit of labour and landless. That means higher demand for housing market.

And second, even if it did work, it would build a general tax-cut bias into your political economy, because every time a wingnut government lowered taxes on real estate, the next responsible government would have to limit itself to revenue-neutral tax reforms in order to prevent a housing panic.

Why would "revenue-neutral" become such a keyword? And land tax is not a "tax" really. It is a return of public wealth to the public.
But of course we have seen this in the USA. They have made there for 100 years politics with "property tax relief" agenda. And property tax is the most hated tax everywhere.