The Apple TV app keeps a Library, which is much like a database of content information (meta data), including file locations for what was imported into this Library. The Apple TV app doesn’t search a media folder.
The Apple TV app by default copies media items on import, to the TV Media folder location set in the app’s Settings. That would default to the boot drive, but can be changed to a folder on an external drive. You can also turn the copying part off, so that files are just linked from where they were. By not copying, some media files may get “missing” if the source file was on a volume that is later unmounted. By taking away some automation, you have to organize carefully for yourself.
My advice for your type of setup is to have the Library database (TV Library.tvlibrary) on the Mac’s internal drive, in your home folder. The Mac should be able to find a Library, even if the external drive is not connected, or else it will create a new empty library.
With that set, you can easily have the TV Media folder location elsewhere, e.g. on an external drive, by changing the settings for both the media folder and the copy-on-import setting.
If you keep the folder structure basically the same on the external drive as on the boot drive, then the Apple TV app can locate more missing files at once, from re-locating one.
Else you may need to locate them one at a time, or remove and import media files again, to fix the situation that was.