Full text search installed or not
My answer:
If FULLTEXTSERVICEPROPERTY says it's not installed, then I would install from the original media. Run through the installer and simply add Full Text Search from the features page.
FTS is fully in the SQL Engine in 2008 so if it thinks it isn't installed, then ya best make it happy.
My opinions/ponderings:
Did you move a database from a previous SQL installation that had full text installed? That might explain the row in sys.fulltext_catalogs.
When you open a Database in SSMS, under the Storage Folder, Full Text Catalog folder do you have the option to add a New Catalog when you right click?
In SQL Configuration Manager do you see the Full Text Daemon Launcher service?
Easy way
SELECT
CASE FULLTEXTSERVICEPROPERTY('IsFullTextInstalled')
WHEN 1 THEN 'Full-Text installed.'
ELSE 'Full-Text is NOT installed.'
END;
inspired by https://www.sqlshack.com/hands-full-text-search-sql-server/