multi column index order for order by
No, your given index cannot be used for the ORDER BY.  You can see this by doing set enable_sort=off and then explaining the query, you get a sort anyway because that is the only way PostgreSQL can get the job done.

But if you create the correct index:

CREATE INDEX ON message_log (website,  in_use, insert_timestamp NULLS FIRST);


then it will be used for the ORDER BY.