I know that Upwork's stand is that google manages search results and its up to them to update. However shouldnt upwork atleast return a 404 when job is deleted, or say that it was deleted instead of saying its private to up work freelancers ?
Are you referring to the job post that you recently canceled? I have already requested Google to de-index the link from their search results. In the future, if you need to remove links from Google's search results, you may refer to this Google help article for more information.
I would also like to note that since this job is private, only users who have access to the job post will see a notification that the job is canceled.