Ah yes, there’s council tax - but tenants pay that too, so it’s not a difference between the scenarios. Also I wouldn’t want or expect to stop contributing to local services just because I don’t have a mortgage!
Tenants in the USA do not pay it directly but do so effectively by proxy, because they pay rent to the landlord and the landlord pays property taxes to local government.
Property taxes in the USA go towards local government services like garbage collection and street maintenance, just like council rates. So I believe property tax and council rates are still analogous.