Solved. I had both an htaccess.txt and .htaccess in the site I was cloning. The site worked fine, but the clone had no admin template. I removed .htaccess in the site, cloned the site, and the clone worked.
In the Joomla Global Configuration, there is a Use URL Rewriting option. In the help text, it says "Apache and Litespeed: Rename htaccess.txt to .htaccess". My host uses Lightspeed. So I turned on URL Rewriting and renamed it. Somehow ended up with both files later on - unclear how. Either way, by keeping htaccess.txt only and not renaming, everything seems to work.