@rachelmcr-you're welcome, even though I didn't do anything other than prod fredyrouge to continue poking around his settings. You should probably reopen his closed thread and note his solution. :) Oh, I see that someone did already reopen that thread. Posted!
After flickr API changes and their move to using only https, it's a shame that something that has been working since oembed was added to WordPress core is now broken. I hope that this will eventually be addressed by WordPress.
Compared to the WPcom built-in slideshow, the biggest advantages of the flickr slideshow are the option to click through to a full-screen slideshow and the line of thumbnail images which can be seen when hovering over the slideshow display which allows a viewer to skip around rather than needing to view many images consecutively (and sometimes very slowly).