Time Machine can't unmount new external HD to initiate backups

How is the disk currently formatted? Is it NTFS? Is it APFS? You may need to try formatting it again like so:

Drive Preparation - El Capitan or Later

1. Open Disk Utility in the Utilities' folder.
2. After Disk Utility loads select the drive (out-dented entry with the mfgr.'s ID and size) from the side list. 
3. Click on the Erase button in the Disk Utility toolbar. A panel should drop down.
4. In the drop down panel set the partition scheme to GUID. Set the Format type to Mac OS Extended (Journaled.)
5. Click on the Apply button and click on the Done button when it is activated.
6. Quit Disk Utility.

The ExFAT formatting is the issue. That's a Windows format and Time Machine needs Mac OS Extended (Journaled) as Kappy says. So yes, you need to erase and reformat with Disk Utility. We have no idea how you came up with "ExFAT )per Catalina" as that's totally wrong.

Do not use that format for Time Machine. Use the procedure I posted which will produce the desired results for using the disk with Time Machine.

See my previous reply.