ex Fat formatted hard drive back up

Big Sur should b able to recognize a drive formatted ExFAT. However, it can't be backed up with Time Machine or used for a Time Machine backup drive.

With the drive connected to your Mac launch Disk Utility and see if it recognizes the disk.

While is Disk Utilities >> View >> View ALL attached Drives and Choose the Upper mOst Name of This Specific Drive ie. WD or Samsung etc. Does it still appear as inactive ? If so, if you what to still use it in exFat - perhaps it needs to be reformatted as exFat with Master Boot Record. If you want it to be used only on macOS as a Time Machine Backup - it needs to reformatted as APFS and GUID Partition Map.