You've only shown dollars having no value for one person; this is not true of all people.
For example: property tax. Not everyone pays it, but many must. These people need dollars, so they are willing to do stuff for them. The stuff that they are willing to do provides the value of the dollar -- regardless of what the non-taxpayers think or do.
For example: property tax. Not everyone pays it, but many must. These people need dollars, so they are willing to do stuff for them. The stuff that they are willing to do provides the value of the dollar -- regardless of what the non-taxpayers think or do.