I struggle with this, too.
Yes, we all know that Reagan beat 'Scoop' Jackson in '76 and then struggled with the '79 energy crisis and stagflation. And we all know that Carter won in '80 and was matter-of-fact about his evangelical views. And was a steady eddie type when his country needed him, and in fact had high approval numbers when his second term ended Jan. 20, 1989.
Yes, it slows down the beginning of the thread to point out all these obvious facts. But for some of our younger members here at Alternate History, I think it might help. Might help remind them of facts of which they're just barely aware. And perhaps on balance worth it.
FOR: The youngsters
Yeah, Carter is considered as one of the best US Presidents in history.
The economy stabilized with him (Volcker had raised interest rates before, so inflation largely ended when he took office). Stable growth rates of 4%, peaking at 6% in 1984, were achieved, and the poor, working class and middle class greatly benefited, as well as the non-multi-millionaire upper class. The MSMEs began their boom during Carter's term.
The Welfare System was also reformed, with GMI enacted.
Environmentalism also became mainstream and numerous environmentalist laws were enacted. Vice President Jackson and first-term SecState Edmund Muskie were so important in this aspect. The clean energy revolution also began under Carter.
HSR and efficient travelling also saw its foundations during Carter, and the Space Program also benefited, with America landing on the Moon again by 2027.
And then universal healthcare. After decades of advocacy by economic progressives, it finally passed in 1985. His being a Southerner and from the working class was a coup for UHC advocates, and adopted a system similar to the French one.
He also repealed Taft-Hartley and in his plan to protect the middle class and workers and farmers, he was pro-union.
But he also improved business policies, making it easier to do business in the US. It's like the "Main Street and Wall Street work together for everyone's good" thing.
Importantly,
he and his administration thought the world how to handle globalization. He advocated that while fair and open trade is fine, it must be done in a way that will still support downscale people. With the Department of Education's founding in 1983 and welfare reform combined with legislative successes in the perod 1981-82, he was able to push for CAFTA and NAFTA while supporting the poor with "hand-up" welfare instead of "hand-out" welfare (according to Forbes*) proper education and skills training and healthcare, thus ensuring the poor thrive in a competitive world.
He also upheld the Fairness Doctrine. That may have prevented media polarization.
Carter restored public trust in the government after the quadruple disasters of Watergate, the First Oil Crisis, the 1979 Iran and Oil Crisis and The 1980 recession. Many people aren't that afraid of government action because of him, according to a new Gallup*2 poll in 1984, 64% of Americans trusted the government, a complete turn-around from the 32% recorded in 1980.
In social issues, he pushed for an abortion ban and curbed pro-gay media*3 especially after AIDS was spotted in the mid-1980s. However, he still pushed for a plan for during diseases, including AIDS, but he discouraged anything related to homosexuality and LGBT rights And because the Democrats are the majority party today, too and because of conservative American SCOTUS justices, social conservatism is still mainstream now, especially since it's connected with socioeconomic justice within the Democratic Party, which signed Civil Rights, after all.
On foreign policy, he did enact a defense build-up to counter the Soviets in the early 1980s, but did reach out to Gorbachev and is seen as the one who ended the Cold War. So in way is he so dovish. He's also pro-Israel, too.*4 Though VP Jackson also influenced him on foreign policy as well.
All in all, Carter's Presidency is considered as in of the greatest. He was then replaced by Al Gore. Jackson died in 1984, and he was replaced by Bill Clinton. However, Clinton did not run in 1988 due to sickness and because he said he wanted to see Chelsea grow up.
His presidency is called the "Carter Revolution", and it marked a shift to economic progressivism and social conservatism and moderate foreign policy (but pro-Israel). Some even say he's the second FDR.
Gore then lost to Bob Dole in 1992, but the GOP got crushed again during the 2000 Great Recession*5. Bill Clinton then became President, and Romney was next. Since 2002, America is enjoying it's longest and most uninterrupted economic expansion in history, and many Americans do thank him for it, according to a Gallup poll*6 in 2015 showing that 74% view Carter's term positively.
Part of supplanting the religious left with the religious right would also require a change in the Republican Party, possibly also during the Seventies. The rise of the Post-Rockefeller libertarian movement and emphasis of that really stunted any long term implementation of the Southern Strategy and the fact that after Carter the religious right had very little hope to foster in the Republican Party. I'd say anything after President Weld would be too far of a stretch.
I'm sorry, but I think you typed the wrong name, as Hillary's our President today*7.
And I think I've established the facts for my fellow youngsters here in AH.com
P.S. The Democratic supermajorities in the Senate and House after 1980 enabled him to enact much of his legislative agenda.
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*OOC: Fictional. Just playin' safe
*2OOC: Fictional as well
*3OOC: I think any religious left or right will be socially conservative. It's just that they differ on economic issues.
*4OOC: Many in the Religious Left were pro-Israel. Christian Evangelicanism is mostly pro-Israel, too IMO. And from what I see in the Internet, he only became pro-Palestine later in his OTL life. And I think there were and are many pro-Israel Religous Leftists in history.
And Christian evangelicanism is also rooted in some form of intervention, though human rights are at the forefront.
*5 Let's say butterflies delay the Asian Financial Crisis and the Japanse Housing Bubble collapse to 2000, joining the Dotcom Bubble in making the economy crash.
*6 Fictional, too.
*7 Eh, they adopt the ATL's Democratic Party stance: Social Conservatism, economic progressivism. Many do change their lives in various ATLs

After all, they were already Democrats in 1980, and if the Evangelicals are in the Democratic Party, they will of course suit their needs
IMO, this is a scenario where maybe America's Christians are not hyper-capitalistic.