I dug a little deeper into the differences between MATE and Xfce. If I run synaptic-pkexec as myself from a shell in a terminal emulator, it works with MATE: I get a pop-up that prompts for the root password, and I eventually get Synaptic.
If I ran it with Xfce, I didn't get a pop-up. And entering the root password didn't work. $ synaptic-pkexec ==== AUTHENTICATING FOR com.ubuntu.pkexec.synaptic === Authentication is required to run the Synaptic Package Manager Authenticating as: root Password: polkit-agent-helper-1: error response to PolicyKit daemon: GDBus.Error:org.freedesktop.PolicyKit1.Error.Failed: No session for cookie ==== AUTHENTICATION FAILED === Error executing command as another user: Not authorized This incident has been reported. $ The "no session" error led me to wonder if there was a daemon missing. I installed policykit-1-gnome, logged out, and then logged in again. And now I can start Synaptic from the Xfce main menu.