Suppose we were to ignore Chinese opinion, and only seek the approval of our friends in the League of Democracies before undertaking coercive action. With the threat of being invaded or sanctioned by democracies, the dictatorships of the world wouldn't just shrug and plod along one by one in their oppressive way. Faced with an external threat, they'd form alliances against all those mean democracies that keep invading and sanctioning them. Nuclear proliferation is a particularly big problem here -- what's to stop China from selling nuclear weapons to dictators who promise to keep selling it oil or promise not to sanction it? As Yglesias points out, there's a serious danger that you'd end up in a great power conflict like the Cold War, with its Vietnam-style quagmires, huge sums of money wasted on armaments, brutal proxy conflicts that rip apart Africa and Latin America, and possibility of nuclear war.
That, obviously, isn't where we want to go. It's not even clear that this kind of polarized situation would be good on democracy-promotion grounds -- when your undemocratic country finds itself threatened with invasion by democracies, being a local democracy activist might make you look suspicious and unpatriotic. So as annoying as it is to look for Russia and China's approval on stuff, it's worth some restrictions on our freedom of action to avoid a situation where the world is embroiled in decades-long negative-sum great power conflicts.
A lot of this is moot right now -- after the Iraq War, our fellow democracies aren't exactly itching to follow the United States into battle. But building a substantial 'League of Democracies' that wields massive power and excludes countries we don't like isn't even really a goal to build towards. Truly inclusive organizations like the UN, even with their obvious drawbacks, are the kinds of international institutions to work through.