I finally called Apple Support and resolved my AppleCare One problem. It is as I suspected: AppleCare One does not correctly handle multiple user accounts in macOS. This is a known issue. No ETA for a fix.
The solution? Cancel AppleCare One and sign up for individual warranties for the devices that were covered under AppleCare One. These new plans will cost me more than AppleCare One and more than I was paying for the plans they had before I signed up for AppleCare One.