"It seems Access (and yes, even dBASE) allowed storage of relevant records in a separate table joined by the main record’s serial number. Didn’t that allow the CareDates themselves to then be filtered for analyses? Does FMP contain a similar process to permit reliable automated analyses of this data?"
Yes! and the fact that you have not done so is what is complicating your databse design.
In Manage | Database, you can define new tables and relational links between them.
You can use a Portal like you would an Access Subform or subreport to list records in a related table.
Look up Portal in the help system to learn more. It should be a familiar concept given your experience with other database apps.