Now I can go to bed.
When success eludes you, always go back and read a tutorial. I did, and it seemed so basic I didn't think much about it. At this point I'm not even sure which version of cgi-lib.pl is there, but it's working.
The fix... Going back to the site's docroot .htaccess file, I noticed that "AddHandler cgi-script .cgi .pl" had been commented out. I'm not even sure it was there on the old server, but I think it had been added when dealing with includes on the home page (some were cgi, some were php). The .htaccess file always did have an "Options +ExecCGI" line and I thought that was sufficient. Obviously not.
So I uncommented the AddHandler and there it was!
Why the logs showed no error on this, I don't know. In any case, even though the solution was outside our discussion, I very much appreciate the time and effort you put in to help. Thanks again.