What I'm looking to do is pass the link juice on from about 5,000 backlinks still pointing at our old site (which exists in domain form only a present) to the new site. Are these links pointed to the home page? Or are these pointed at relevant landing pages? If to the home page (and page is same on new site) nothing is needed. If these links are pointed to a landing page, you will need to route them to a related page on the new site and 301.
Are you saying there's not point in using a 301 if the page doesn't exist? Yes, no point unless there is a new landing page that is some-how related.
I don't really understand why you'd use a 301 if the page still existed...? If the page is on the new site and is the same url, no 301 is needed.