I save them on the expectation that Apple will someday release a better tool for browsing them. Especially the texts between me and my wife, they are a story of our life from the very first one.
When I went to open the email on my new machine, the loading indicator just spun and spun until finally my browser crashed. When viewing an email conversation, Gmail will display previews of all the emails in the chain, including their attachments, which was causing my browser to crash. Next I tried connecting my Google account to Apple Mail, but Gmail's SMTP server would time out before the attachments were able to download.
After a while I forgot about the photos all together, until I randomly stumbled across them while searching my Gmail account for some other files I had sent myself around that time. At this point it was 2014 or so, and my browser was very easily able to open the messages so I could download the photos, which of course at that point I had zero interest in.