I’m going to answer my own question, though I don’t understand.. yet. I’ll try to research further, and would like to see if someone else can explain.
I overwrote the wp-admin/widgets.php file in one of the remote sites – which was created with a Softaculous script – with the same file from my development site, which I created manually, and the problem cleared up.
A file comparison of the widgets.php from the other remote site and the one at my local development site says they are identical. Same result: Uploaded the widgets.php from dev to remote, over-writing the latter, and now everything’s OK again.
Weird.
ADDENDUM
My only guess at this point is that it has something do with character or file encoding on a deep level. I’ve seen similar problems and solutions reported in relation to other files and processes. This is the only example involving widgets.php that I’ve seen in my research.