Picasa embeds have been deprecated so it is recommended that we move to alternatives for slideshows.
Here are some alternative methods.
Someone has posted to the help forum an alternative method that uses YouTube.
You can read their instructions here
Their forum post is HERE
And here are some examples:
Note that you cannot add and remove images to your slide show in YouTube. You can only create a new slide show with the images you want to appear in it.
The only way I have been able to make a slide show auto start is shown here
This involves using the animation feature in Google Photos. The procedure is explained by Peggy K, a top contributor in the Blogger forum. You can find her post about this in Google+. She explains about using it in Blogger but as you can see it also works in Google Sites. You should view her detailed instructions but here is my condensed version:
You cannot change the speed but viewers can click the image to see it larger.
To make a slower animation you can use an editor like LunaPic
You can upload your fast GIF, change the speed, save the image and
upload it to Google Photos, get its URL and insert it into your page.
(Don't forget to move it into a shared album)
Another option is to use Google Slides (Presentations). You can edit them to add/remove images.
See more information on using presentations.