The latest issue of Fortune magazine devotes a full eight pages to explaining how U.S. Senate Minority Leader Mitch McConnell of Kentucky “is fighting to bring back the grand old Grand Old Party.”

If McConnell can fend off his conservative challenger on May 20, guide four of his fellow incumbents through a thicket of Tea Party primary challenges, and survive the November election — a difficult hat trick — he is likely to emerge as the one Washington Republican who can return the Grand Old Party to the grand old political center and put an end to Tea Party extremism.

“Anybody around town who’d like to have a more pro-business, pro-free enterprise government, the only thing they can do about that between now and 2016 is change the Senate,” McConnell says. “Because the one thing virtually every Republican has in common from Maine to Texas is that we are essentially the party of the private sector.”

To hear the McConnell camp tell it, victory would give the senator the chance to forge a new Republican vision — to finally offer those much-promised legislative proposals for economic growth and reform that the party has so far been unwilling or unable to unite around. Defeat, on the other hand, would extend the muddle of the GOP’s internal squabbling and diminish its presidential hopes in 2016 and beyond.

The fact — how do we put this? — that McConnell is not naturally groomed for human contact makes this battle for the soul of the Republican Party all the harder to win. He is better suited for cloakroom fencing than campaign glad-handing and soaring rhetoric. And he’s awkward on the stump, shrugging off the retail exertions most voters expect of their pols. …

… Of all the people who think McConnell can pull off a win, though, no one is more of a true believer than McConnell himself. As he once told an associate, “It’s amazing what you can do when your back’s against the wall.”