Whether a tax is progressive or regressive tends to be quite a fundamental question economists ask. Lots of proponents of a land value tax claim the tax is progressive. So either they think that is a good argument in favour of it (in which case shouldn’t their argument be weakened if the tax is not progressive? Surely one should care about that?) or they think it is immaterial, but then why mention it at all?

Regressive taxes coupled with payments are a very powerful way to squeeze the lower ranks of the middle class, while preserving the rich and the higher middle class.

You can just structure the payments however you like to prevent that.

You can structure the rates however you like to place the squeeze at any place you want. If you structure it to completely get rid of the squeeze point you have just created a proportional or progressive tax.

You've created a progressive transfer system.

Whether those transfer payments are via taxes, payments, or services is not essential for the goal.

If you encode it in tax rates, you are subsidizing behavior that you are trying to deincentivize via taxes, which is unfortunate.

The idea of LVT or other taxes on externalities is to tax behavior you want to prevent. That's why we should tax the capture of land, the pollution of the environment, adding traffic to a transportation system, etc. That's why it's unfortunate to tax things like "making your house nicer" or "converting a parking lot to a veterinarian office" or "hiring another employee". You wouldn't sent checks to someone because they let their house decay or they fired someone, so why would you give them a tax break?

The point of the transfer system is to help people who need help, right? Not to discourage things we want to happen.