Old MacBook Air unable to upload to icloud

First thing we have to know: And how is your external drive formatted? To avoid damaging the Photos Library an external drive must be formatted in either APFS format or Mac OS Extended (Journaled) format. There have been so many problems with using incompatible drives that the newest macOSs won't even allow a Library on a non-Mac formatted drive to open, since there is a chance of damaging the Photos database. See this:

Move your Photos library to save space on your Mac - Apple Support

If this drive is in a an incompatible format, stop running Photos with it immediately!  A Photos Library can sit on an incompatible drive, but running it may corrupt the database.

How much space is there on the internal drive of the Monterey MacAir?

Mac OS Extended (Journaled) format is good!

Check Photos' Settings>General to see if this Library is the System Library:

If it's the System Library, then that should be grayed out like above.

Is it connected to iCloud? Check Settings>iCloud

At the bottom of my Library I see this:

If the Library is connected to iCloud, then it should say something about syncing at the bottom-- I think that's right, but I don't remember Monterey well…