Service crashes when using AppleScript to click menu bar icon
This works for me on the latest version of macOS Sierra.

I followed your steps exactly.  However, this code you posted needed to be tweaked to work on my system correctly.   Here is the code I used and placed into an Automator run AppleScript action

tell application "System Events"
    click menu bar item 3 of menu bar 1 of application process "SystemUIServer"
end tell


I also did get the same error message you got when i tried to run the service from within Finder (after successfully running the script within Automator and ScriptEditor)  

On a hunch, I added Finder app to the list of allowed apps to control my computer. 



Then I went back to Finder services menu and attempted to run the service again.  Bingo! It worked.  I did the same thing with several other applications I had open and it did not produce the error message