Possible reason to include a tsvector field?
As you asked for possible reasons to include a tsvector field, I think one is for sure if you had to put into the tsvector text from columns of other tables.
If you have a table of blog articles and you want execute a full-text search over title, content and related tag names, but tags is a separated table you can't, unfortunately, use generated column or function in index, but you have to update your tsvector (with a trigger or similar) and than you can update an index on the column.
I hope I gave to you a reasonable reason to use a tsvecor field.