For a typical homeowner, the effect of LVT would be pretty neutral, by design. Proposals I've seen advocate changing the _distribution_ of single-family home taxes between land and improvements, but not the _total_.
Yes, LVT would discourage speculators from buying land as an investment without improving it, which is a societal good. But it's not direct and short term like many of your other examples.
I think GP's point stands that rolling back LVT would be highly desired by big landowners (who could then return to do-nothing land appreciation and rents for their profits rather than be required to actually _do something_ with the land to earn over its taxed value), while the average Jane couldn't care less, because it wouldn't really impact her tax bill.
That sort of situation is not stable long term. To make it more stable, you'd need to set it up so that LVT lowered the average voter's tax bill. I'm not sure how feasible that is.