You're right that it does have an effect, however if you look at other areas with a higher property tax, houses are still extremely expensive in desirable areas. It doesn't have as big of an effect as you'd think and it severely penalizes the middle class who are forced to relocate so that the ultra wealthy can take their place.

>The local government will get the money in a different way

This is a feature IMO, the government can get that tax revenue but in a more fair way e.g. income taxes that are skewed higher towards the ultra wealthy rather than just kicking people out of their homes, where they probably do want to live.