Closers are overrated. I know that sounds nuts, but when you consider that the final possessions are as important as the beginning possessions (though the final ones are higher leverage), then you should probably consider that our attachment to "closing" is more emotional than logical.
LeBron James has had some great moments in whatever we consider "the clutch" amid some notable struggles. But I think that his handling of these specific situations is less important than his overall game play.
Let's look to Kobe Bryant, oft known as a "closer," as an example. As Henry Abbott demonstrated in this epic ESPN piece, Bryant's Los Angeles Lakers were nothing special in the clutch, despite the mythology. But, in part due to Kobe's prowess, the Lakers were good enough to win championships with a flagging crunch-time offense.
So the focus shouldn't be on whether Wade or James gets the last shot. Instead, the biggest question should be, "Are the Heat good enough?" Presently, it seems that Miami isn't close to the Spurs in overall team play. This matters more than how clutch LeBron is or isn't, or what he does as a closer.