Is there any way to take a preexisting field with content, and generate taxonomy from it?
It sounds like a job for an entity reference field

For example, add a "field_classes" entityreference field to "Instructor profile" referencing the class nodes that instructor teaches.

Alternatively, add a "field_instructor" entityreference field to the "Classes" content type, referencing the instructor profile(s) for that class.

Using Entity Reference, each content type can easily list the related instructors and classes.