The New York Times on Wednesday carried a provocative front-page dispatch from Kentucky by Abby Goodnough, who’s been doing a superb job of covering the impact of the Affordable Care Act in the Bluegrass State as a window into the law’s import nationally. The piece captures the political dynamic in Kentucky, where Obamacare has slashed the rate of the uninsured from 20.4 percent to 11.9 percent, the second largest drop after Arkansas, but where this success is seemingly not redounding to the benefit of Secretary of State Alison Lundergan Grimes, who is now lagging in the polls against Mitch McConnell. This is indeed a conundrum for Grimes, but I worry that the piece, well-reported as it is, may leave many readers with a slightly mistaken impression—and place the blame for Grimes’ dilemma in the wrong place.

Goodnough quotes Kentuckians who are, themselves or their close family, benefitting from Obamacare, but are nonetheless planning on voting Republican. For instance:

The Affordable Care Act allowed Robin Evans, an eBay warehouse packer earning $9 an hour, to sign up for Medicaid this year. She is being treated for high blood pressure and Graves’ disease, an autoimmune disorder, after years of going uninsured and rarely seeing doctors.

“I’m tickled to death with it,” Ms. Evans, 49, said of her new coverage as she walked around the Kentucky State Fair recently with her daughter, who also qualified for Medicaid under the law. “It’s helped me out a bunch.”

But Ms. Evans scowled at the mention of President Obama—“Nobody don’t care for nobody no more, and I think he’s got a lot to do with that,” she explained—and said she would vote this fall for Senator Mitch McConnell, the Kentucky Republican and minority leader, who is fond of saying the health care law should be “pulled out root and branch.”

No doubt, there are many GOP-inclined voters like Ms. Evans among the several hundred thousand Kentuckians who have obtained coverage under Obamacare. Their existence, and quotes like this, only confirm one of the most popular conceptions of red state politics in the minds of blue state liberals: the “What’s the Matter with Kansas” notion that lower- and middle-class white voters in these states are voting Republican even though their economic interests are far better represented by Democrats. How else to explain that a state like Kentucky could be benefitting so much from the law and yet be leaning Republican?

But this likely takeaway from the piece—like the "What's the Matter with Kansas" thesis more broadly—may oversimplify things. For starters, my strong hunch from my own reporting in the region over the past couple years—including several trips to Kentucky for a new book on McConnell—is that the Democrats’ biggest problem in Appalachia and the Upland South is not that the people who are benefitting from Obamacare or would stand to benefit from it if their states fully implemented the law are voting against their own interests, for Republicans. It is that many of those people are not voting at all.

Remember, the vast majority of Obamacare’s beneficiaries in states like Kentucky that expanded Medicaid are poor, with incomes below 138 percent of the poverty level, the threshold up to which Medicaid coverage is to be expanded under the law. And poor people famously vote at far lower rates than everyone else, with turnout lower in Appalachia than just about anywhere. When I spent a weekend in 2012 in the mountains of eastern Tennessee at a free health clinic full of people who would be covered under the law’s Medicaid expansion, I was expecting a “What’s the Matter with Kansas” type reaction—people who were meant to be the law’s main beneficiaries railing against Obama. Instead, the prevailing attitude toward Obama was far more sympathetic—the clinic attendees generally felt like he was on their side. There were just two problems: they had virtually no inkling of the law’s intended benefits for them. And, even more crucially, most of them said they generally did not bother to vote.

By now, the first of these two problems should have dissipated in Kentucky, where the law has actually gone fully into effect, with glowing results: Low-income voters should be aware of the law’s benefits for them. Yes, there are reports of Kentuckians not realizing that the “kynect” that is covering them is in fact “Obamacare,” a confusion based in the decision by the law’s promoters in Kentucky to give the law a state branding, rather than a pejorative national one. But I have enough faith in people's common sense to doubt that this confusion is all that widespread now, nearly a year into the law’s debut.

No, the bigger problem for Democrats in Kentucky is the second one I identified at that free medical clinic, the political disconnect of the low-income population that stands to benefit most from the law. And this is where the scrutiny really needs to shift away from the voters and to the candidate. Alison Lundergan Grimes’s campaign has made the decision to talk about Obamacare as little as possible, even though the law has had a bigger impact and better implementation in Kentucky than just about anywhere in the country, thanks to the staunch backing of Governor Steve Beshear, a Democrat. Her campaign is avoiding the issue despite the fact that McConnell is so aware of his vulnerability on it—vowing to repeal a law that has provided coverage to hundreds of thousands in his state—that he has wound himself into comical pretzels to obscure the ramifications of his stance, claiming at points that “kynect” could survive even if Obamacare is repealed. The Grimes campaign briefly pounced on this blatant disingenuousness, but has not been nearly as aggressive pressing the point as it could have been.

Evidently, the Grimes camp worries that talking about Obamacare’s benefits for Kentucky necessarily yokes her closer to the law’s namesake, who is not popular in the state and who is constantly named in the barrage of ads that have been attacking Obamacare, virtually unchallenged, across Kentucky. They assume that talking up the law will inevitably backfire with the likes of Robin Evans. But deeply anti-Obama voters like Evans are likely a lost cause for Grimes regardless. What her campaign could dearly use is higher turnout from Democratic-inclined voters who, like many Kentuckians, have a low opinion of McConnell and who haven’t voted in the past but might just be prompted to do so with a strong pitch for the benefits of “kynect.” Grimes, to her credit, has tried to reach out to such disconnected, downscale voters with a sustained message in favor of raising the minimum wage. But why not strengthen that "I'm with the little guy" appeal even further with a forthright pro-kynect message that both appeals to self interest and highlights McConnell’s double-talk on health care?

Over in Arkansas, Democrat Mark Pryor has been talking up the law, albeit in veiled terms, and, it should be noted, is closer in the polls than Grimes is. Her campaign’s decision not to follow his lead is giving it an aura of trepidation that is never good for a campaign to have, and is especially unfortunate for a candidate who has, in other contexts, been so impressively bold in challenging McConnell. As Al Cross, a longtime political reporter with the Louisville Courier-Journal now at the University of Kentucky, wrote over the weekend:

McConnell has never been made to explain how Obamacare could be repealed "root and branch" without affecting kynect, the state health-insurance exchange. So far, this race has been almost like the "Seinfeld" TV show, which was famously about nothing but the characters. The unpopular McConnell has made it about a third, even less popular character: Obama. Grimes, who is still a fuzzy figure to many voters, needs to make it about something else.

The Grimes campaign has shown it knows how to fire a shot. If it’s unable to get more voters whose health care McConnell wants to take away to turn out against him in November, the fault belongs to the campaign, not the voters.