Capturing that gold is a fools errand.
Destroying the two gold mines in Virginia City and Colorado can be done against a human. You need to get partisans with the RGD cards. Partisans have the pillager trait and will auto destroy the mines if they take the region. Though, they will only destroy the mines if the region is loyal to the USA. So don't play demonstration cards on the regions to increase their loyalty, normally one would play those if they wish to capture the mines with cavalry and irregulars.
To hit Golden City, CO, partisans can be recruited in two regions to the south... In South Park.
For Virginia City you can send partisans up from Western Arizona and the two Southern California regions - it may not be obvious that those regions are adjacent to Virginia City.
The icons of the gold mines on the map won't be removed when they are destroyed. Check the tool-tip for the city to see if the mine is present. You will also get a note in the message log when the mine gets toasted.