I think the main advantage is really that gate staff finds it easier to check and therefore is actually enforcing it.
Before/in other alliances they are sometime rather hesitant because they are confronted with all sort "rules" (aka excuses) where they, as gate staff, can only lose: Either they don't enforce and upset all those really entitled or they do enforce rather strictly and risk a complaint from someone who's entitled but sent back. SkyTeam priority - together with the easy to spot logo - gets them the confidence they need.