You do you. This has different access than the roads surrounding and should not be marked the same. So access=private here has a different meaning from access=private next to it.

The map is called “WSDOT - Bike Paths Along State Routes” at a WSDOT.maps.arcgis.com link. The sign you linked doesn’t appear to be physically there any more. A lot of those have been taken down and burned (seriously). You could still be correct that it’s the wrong name as the line is shown as city owned and the map is built by the state.

The old name is A St. this bit of road is not accessible from Barnes and is not on McChord, it’s on Fort Lewis. Someone should never have stuck Barnes’ name on this section. Probably just due to signage for Lewis’ trail roads lacking.