@Will.S I don’t believe this is possible currently (i.e. you aren’t off base )
I assume the custom field added to the templates wasn’t a global field (part of the library) so each of the projects created from the template now have their own “version” of a project-specific field now unrelated to one another. As you are likely experiencing, it can be quite the lift to manually correct this issue where you have to add the global field and copy the data over.
It is a lot less time consuming with dropdowns since you can sort by the current field and then perform a bulk select change. line by line for text and number fields unless you see a lot of similarities in the values where the sort will work the same way.
You got my vote for a more automated way to “convert” or “merge” from one field to another!