I would suggest that you are trying to schedule a content item or page that is already published. In this scenario the desired content will be marked as scheduled but it will appear, because it is already published. In this case you have to unpublish the content first and then you can schedule it without any problems.
Could you please then recommend the best way to schedule an update to an existing page? For example, an existing page will have new content added to it, but that content should only show up after a certain date.