When you give the COLLATE NOCASE DESC option in the index it means convert all the text to lower case and store them in descending order for case insensitive search. So, to make use of that index, you have to give the same collation again in your query, and the following should work.
SELECT profile_id FROM profile INDEXED BY idx_name WHERE name == ? COL
If you recreated the index without this collation, then it means you cannot do case insensitive search, or have to convert things to upper or lower case every time like below.
SELECT profile_id FROM profile INDEXED BY idx_name WHERE lower(name) =
Although you have found a workaround, this may be useful for someone else, facing a similar problem.