The problem is that the slideshow is not mobile friendly, so it can only be viewed on a desktop browser.
So, how do you embed a mobile-friendly Picasa/Google+ photo slideshow to Google Sites? Here is a step-by-step tutorial using an RSS feed:
1. On your Google Site, click on Insert and then More gadgets.
2. Search for "image slideshow rss", and select the first gadget called "Image Slideshow"
3. Go to the "old" Picasa Web Albums and choose your album (does not work if you go to Google+ Photos)
4. Make sure your album is public on the web or shared with anyone with the link. Then, right-click on the RSS feed link and copy the link location.
5. Go back to your gadget and set up the settings like so:
And there you go!