Check your steam controller settings to see if playstation and/or xbox controller compatibility is turned "on" or "off" in settings. You can get at this setting only in big screen mode of steam, under controller configuration. I had a similar issue with my steam controller when Steam switched that PlayStation compatibility to "on". I imagine it could also happen in reverse, where it needs to be on rather than off for your specific configuration because how it sets changes how the game recognizes the controller.
Also check if you have any errors in the BindingsLoadingErrors.log file that can be found in the directory "C:\Users\<username>\AppData\Local\Frontier Developments\Elite Dangerous\Options\Bindings" directory. Any errors may need to be resolved before the controller is recognized again.