Select a menu item in AppleScript without using "System Events" in 10.9 Mavericks
I don't know how to click menu items without using System Events, but if others get here searching for how to click menu items with System Events:

tell application "System Events" to tell process "Finder"
    click menu item "New Finder Window" of menu 1 of menu bar item "File" of menu bar 1
end tell




tell application "System Events" to tell process "Finder"
    set frontmost to true
    click (menu item 1 where its name starts with "Compress") of menu 1 of menu bar item "File" of menu bar 1
end tell




tell application "System Events" to tell process "Finder"
    set frontmost to true
    tell menu bar item "File" of menu bar 1
        click
        click menu item "Open With" of menu 1
    end tell
end tell


You can allow all currently installed applications to use the accessibility API by dragging them to the list in System Preferences: