No matter the type of tax (LVT, income, wealth, carbon, etc…) the money goes to the same place: the government.

From there, hopefully the government uses it wisely in ways that redistribute wealth and stimulate growth, but that is orthogonal to how the money is raised.

So instead... What? The high price right now to live near work or relocate is regressive. I don't know what proof there is that "money not going back in to the economy"* is a larger effect than increased housing affordability?

*btw, if the money went toward interest on debt, all else being equal, that is money back in to the future economy via reduced tax demand. Of course this won't actually happen and the government will spend every cent it gets and more. So maybe your opposition should be with the government and not LVT?

How does that not go back into the economy?

The big difference between holding land vs gold/stock is that land is finite. If the price of gold spikes then people will build more gold mines. That's economic activity. If the price of land goes up then people can't make more so there's no economic activity.

If the price of stocks go up then more companies will IPO which directly funds economic activity (those companies's operations).

Granted, it doesn't as well into the economy as directly as if it were tax'd and then spent improving bridges and whatnot.