I haven't done this in a while, but you might want to check out the community controller configs. I had a 3rd party controller with a few buttons that steam couldn't detect, but someone had uploaded a config that worked perfectly for it.

Try going into the Steam settings and make sure that the appropriate controller is enabled for steam input. I had the same issue recently when switching from an Xbox controller to a Gulikit. Enabling Xbox input in Steam fixed it - even though the new controller is not an Xbox controller, and a real Xbox controller worked fine without enabling it.

Can’t say for sure it’ll meet your needs and work with your Logi gear, but I use AntiMicroX to re-bind unrecognized controls.