There is a reason that deadlines exist.
While your agent, editor, or publisher might set them to ensure that a project does not drag on and on … and on, there are probably even more likely solid business reasons behind almost all of them. If your editor informs you that final comments must be received by a certain date, that is almost certainly because, if an official launch date has been announced for your book, there are specific and fixed dates by which your publisher must deliver items like final print-ready covers, interior content, and other materials.
These dates are inflexible, and large companies are not going to bend their rules because the author wants “just one more small change” on the day before these items are due. Remember, publishing is a business and we are forced to play by their rules. That is the reality of this and many other industries. (If you don’t want to follow those rules, or feel that they should not apply to you, then self-publish. And best of luck to you.)
You might get by with it the first time, particularly if the publisher has already invested too much time, money, and other resources into your book to pull out now. But they are going to be a lot less likely to want to pick up any additional books from you or be as flexible the next time. Don’t be surprised if you see new penalty or schedule enforcement clauses in your next contract.
As an author, you might not understand — or care to know — all of the moving parts required to get your book out into the marketplace, but at least respect the time and deadlines given to you by your editor and publisher. Your career will thank you.