including an external drive's contents in Time Machine backup

In the example below, External SSD is excluded. Normally, to remove an exclusion, you select the volume and click on the "-" button. I have not seen a case where a volume is greyed-out except for Time Machine volumes themselves. That said, all my volumes are APFS.

- Pie Lover

P.S. The terminal command tmutil isexcluded may be useful

I appears you tried to add a screenshot to your question. None is present.

To add it to a reply

Click on the add image icon right below the test entry box

After clicking on the icon you are present with a Finder like windows. maneuver to find the image file you want to post and click on it and select Open in bottom right or double click on the image file to add the image in the test reply box.

How is the drive formatted? Time Machine can only back up disks formatted with a macOS format (Mac OS Extended or APFS).

If it is a Mac format, was it used as a Time Machine drive?