Short Answer: Yes, based on the electrochemical/reactivity series, it is possible to percipitate gold from solution using metals higher on the series like copper or zinc. However, it isn't done often because gold is highly unreactive, and when gold salts are formed, they are formed deliberately under atypical conditions like very high temperatures or acidities.
It would be possible to displace gold from it's ionic salt (gold (iii) chloride) using zinc or copper. This is because, as you mentioned copper is higher on the activity series than gold.
The likely reason to why companies don't use single-displacement reactions to retreive gold from solution is be cause gold salts rarely exist. Gold is extremely unreactive (see here) Gold (III) chloride is normally synthesized by reacting gold metal with chlorine gas at 180 degrees Celsius. This temperature is higher than anywhere you'd find on Earth's surface, and chlorine gas is too reactive to exist in that state in the atmosphere. These conditions are typically found only in the lab and not out in nature. Another example of a man-made gold salt is aqua regia (a 3:1 ratio of hydrochloric acid and nitric acid) which can dissolve gold. To recover the gold, ions are displaced using ferrous sulfphate reducing agent (source).
If you are interested in gold recovery, here is an excellent article on a variety of methods.