Yes, pg_trgm was built for that but someone figured out how LIKE could be hacked to use the trigram indexes (gist_trgm_ops, gin_trgm_ops). So if you have a * _trgm_ops index on the column normal LIKE and ILIKE queries may use that index.
I assume your example would be a false index hit which then is necessary to verify against the real value. The same would apply to make sure 'This' is not a hit when doing a case sensitive search for '%this%'. So index-only LIKE scans are not possible with * _trgm_ops indexes.
EDIT: I just realized how awesome the extensibility of PostgreSQL is. An extension can make a core operator such as LIKE indexable in an entirely new way without touching the core code.
I assume your example would be a false index hit which then is necessary to verify against the real value. The same would apply to make sure 'This' is not a hit when doing a case sensitive search for '%this%'. So index-only LIKE scans are not possible with * _trgm_ops indexes.
EDIT: I just realized how awesome the extensibility of PostgreSQL is. An extension can make a core operator such as LIKE indexable in an entirely new way without touching the core code.