However, whenever I run the Live Update program, it re-creates the KJFK folder listed above, and re-downloads all of the scenery texture files that I previously deleted.
Maybe you Uninstalled JFK from the Windows Control Panel after it was *migrated* from an old installation ? In that case, the Uninstaller didn't really worked, because it's basically not possible to retrofit an uninstaller after the file it originally installed were moved. This is a side effect of the migration process but, the alternative would have been forcing all users to uninstall everything, redownload everything and reinstall everything.
To do a proper removal, in addition to the \Addon Manager\Fsdreamteam\KJFK folder, you must remove the following folders too:
Addon Manager\Simobjects\FSDT_KJFK
Addon Manager\Couatl\JFK2
Of course, this won't be required if you originally installed it using the updated installer but, if the scenery was migrated from a previous installation, the above folders must be removed manually, otherwise the Live Update will think the scenery is installed, and will try to restore its files.