I think this will not apply, and likely theme related, but I will ask to receive confirmation. In the “Advanced TOols” there is the powerful option to search and replace content within the database; is it possible for a theme to hard-code links within a database which may be preventing some links on the site from updating to a secured link (https), and if so, use this tool to replace these links?
It is possible that there are full links stored in the database (rather than using the stored site URL).
You can use the UpdraftPlus Search/Replace tool to edit any instance of a URL in the database.
However, I would heavily advise that you first take a backup of the database, and be very precise in what you enter into the tool (as a mistake can break your database).