Safari in Beta 4 unable to open private window

Private windows work just fine for me.

Try deleting all your Safari settings (~/Library/Preferences/com.apple.Safari* and ~/Library/Safari … note that if System Integrity Protection is enabled then you'll have to do it with the Finder as command line progarms can't touch the ~/Library/Safari directory)

Or download the Safari Technology Preview and try using that. (It has separate preferences from Safari.)