SQLSERVER FullText search troubleshooting
Whilst trying to re-create the problem with a fresh table in a separate DB, I ended up comparing the Full-Text catalogue settings. In the DB where I saw the erroneous behaviour the Table Full-Text Pending Changes value was 21219 (!).

I have then checked the FT log and found that the Asset table hasn't been updated for a long time:


  An internal query to load data for a crawl on database '-' and table
  'Asset' failed with error code 1205. Check the sql error code for more
  information about the condition causing this failure. The crawl needs
  to be restarted after this condition is removed.


and also


  Error: 30059, Severity: 16, State: 1
  A fatal error occurred during a full-text population and caused the population to be cancelled.
  Population type is: AUTO; database name is Media (id: 6); catalog name
  is - (id: 6); table name Asset (id: 1076966963). Fix
  the errors that are logged in the full-text crawl log. Then, resume
  the population. The basic Transact-SQL syntax for this is: ALTER
  FULLTEXT INDEX ON table_name RESUME POPULATION."


I had to manually resume the FT updates. After that the search has returned 3 items!