YouTube – How to Disable ‘Are You Still Here?’ Prompt in Playlists

The playlist is available in the current design (One Channel, which you're referring to as "new design") if the user has featured a "newest video" shelf on their channel page – click the "play all" button.

In the now-new design (Material Design), there is a link to play all videos on the /videos tab of any channel.

If you're using Gmail, you can create a filter so these messages don't hit your inbox. If you don't want any emails from YouTube, you can search for it and then click Create filter. You can add additional criteria on the next page, confirming the results before creating the filter.