Cover Art on iCloud Missing in Finder

What did you try to remove ID3v1 tags and which tag version do you write to the files at all?

See the documentation on tag types:

To remove all tag type that are not ID3V2: tick remove for all except ID3V2.
Then select the files and apply "Cut tag".
To save only ID3V2: set write only for ID3V2.

How did you try to save ALBUMARTIST? if you do not have the field in the tag panel and/or the file list, then you can at least add data for it via the extended tags dialogue.

I successfully removed IDv1 tags and added ALBUMARTIST to 2 files on the Desktop.

After copy/paste the files to iCloud/Downloads folder, the album art disappeared.

I think that this indicates that it is not a problem of the presence of ID1 tags.
The display of the ALBUMARTIST would have been another test to see whether the V1 or V2 tags are being read. If ALBUMARTIST is shown then this would indicate that V2 tags are read as V1 does not have an ALBUMARTIST.

The behaviour of the Mac programs is now a problem of the OS but not an MP3tag or tag problem.