You can definitely make ginger beer with wild yeast (fresh ginger root conveniently comes with lots of wild yeasts) and make delicious drinks. There are lots of recipes.
I also recommend, if anyone is interested, investigating “ginger beer plant” (GBP), which is a symbiotic culture of bacteria and yeasts (SCOBY) such as that used to brew kombucha and kefir. It was once common in Britain, but almost went extinct. Some was saved by a German culture bank (DSMZ), and now you can buy some online! Makes a different, and I think delicious, drink compared to the “ginger bug” method!
You don't have to ferment it to get a rich flavor. You just have to use fresh ginger instead of dried or powdered. We make it with Fresh Ginger, Lemon juice, orange juice, cane sugar and molasses. Then force carb.