and no matter what I do, I only get May and June 2013 showing up on my new system.
Sorry to say, but that is exactly how it is supposed to work.
Your Photo Stream syncs with what is currently stored in iCloud and not with what is shown in the Photo Streams on other devices. iCloud will hold only the last month of photos, or the last 1000 photos send to the stream, and then the photos will be deleted from iCloud. Your devices will keep, what they downloded from the stream, but when you sign in to PhotoStream with a new device, that will only see what is currently still in iCloud.
Your device will keep up to 1000 photos, even if they are older than 30 days. Photos older than 30 days are removed from the iCloud server, so all devices may not have the same photos, depending on when you enabled Photo Stream.