This worked. There was a single file, object-cache.php, in the wp-content folder, and when that was deleted, the Media Library started working again.
Weeks of frustration over at last…. :-D
I tried the 2014 WP theme and still had the same problem with the media library loading.
One possibility: We have an error message that reads: W3 Total Cache Error: some files appear to be missing or out of place. Please re-install plugin or remove /home/[user name]/public_html/wp-content/object-cache.php. We have deleted that plugin but the error message still appears. Could that have anything to do with it?
I’m sorry I didn’t respond sooner, Josue. Thank you for your fast reply.
To answer your question, I’m a little hesitant to change to the 2013 / 2014 WP theme — if I do that, is it going to wipe out all my settings in Enfold? I would hate to put it into the default theme and then not be able to get my website back when I restore Enfold.