Thanks for your reply. My experience was that doing the upgrade from 2.1.1 all the way through 2.2.x didn't ask me to do a db upgrade. By that I mean I could upgrade, go to the moodle domain name, and get a login page. When I went to 2.3.x or 2.4.1, going to the domain name redirected me to /admin, and - without prompting for a login or password - told me a db upgrade was required. It would let me, not having an admin password, proceed through the check page, where I got all OKs except for the moodle version. Whether it would have let me actually do the db upgrade w/o knowing the admin password I don't know since I never got that last required OK.
Through 2.2.x I was ok doing the upgrade with Installatron. After that, again just going to the customer's moodle domain name post-code-update, I'd get a warning that somebody (Installatron) had copied this new moodle on top of an older one, so I'd do the recommended moving the directory, copying in fresh code from moodle.org, etc. And then get the db upgrade request by just visiting the customer's moodle domain name, again without knowing the admin password (the main moodle domain would redirect to /admin automatically, but not to a login prompt.)
Question: is it more practical to export the (3) courses from 2.1.x, make a fresh 2.4.1 install, and import them? Would other data from the old db be required besides what the import would provide? I've never exported/imported anything in moodle before, but would kind of like to get as clean a start as possible at 2.4.1. Does the course import/export include all the required user account information, for example? Or is it better to continue trying the code updates followed by trying to do the db verify/update manually?