There are two major issues here. The first, as discussed above, is that voters tend to be suspicuous of intellectuals, especially in the modern era. However, this can be overcomeThe second is that academia isn't really good preparation for a career in politics. The average PhD
as of 2012 graduated at the age of 33, and you would need another five to seven years after that to earn tenure, so you'd be almost 40 before you could think about entering local politics (historically, PhD's didn't take as long to earn, but it would still be a longer time investment than going to law school or into business).
If you define a scholar/intellectual as someone who's published peer-reviewed journal articles or monographs, you could meet the challenge by slightly tweaking the career paths of OTL politicians. Clinton and Obama could both qualify by writing articles for law journals during their jobs as law professors. Ford and Nixon also earned law degrees, although their careers would be less likely to include academic publishing. Wikipedia lists Jimmy Carter as completing a "postgraduate physics course program" at Union College, so a POD could be him deciding to earn a doctorate in physics after leaving the Navy rather than running the peanut farm.
You could also consider PODs or future timelines for politicians with academic backgrounds. These might include:
Henry Cabot Lodge Sr.: Earned a PhD from Harvard and was a member of the American Antiquarian Society and American Academy of Sciences and on the board of directors of the Smithsonian Institution. PODs could include being selected to succeed Theodore Roosevelt in 1908 or running in 1920.
Paul Douglas: A professor of economics at the University of Chicago before being elected to the Senate from Illinois, was seen as a potential candidate for president in 1952 but refused to run. A POD could include him deciding to run in 1960 or JFK selecting him over Johnson as a running mate (you would need the South to be less contested in 1960 for this to work).
George McGovern: A history professor and the Democratic nominee in 1972, but it would be very difficult for him to get elected without either substantially changing his political position or Watergate breaking during the campaign.
Daniel Patrick Moynihan: PhD in history and professor at Syracuse University before entering politics, but it's difficult to see him having a chance at the White House
Phil Gramm: Professor of economics before becoming a senator from Texas, ran for President in 1996, but his candidacy was derailed by a scandal over his investments in pornographic movies.
Newt Gingrich:Was a professor at the University of West Georgia before running for Congress. Ran for president in 2012, but a more plausible timeline would have him running in 1996 or 2000, or some sort of 25th Amendment scenario where Clinton and Gore are both taken out in the 1990s.
Paul Wellstone: Professor at Carleton College before running for senate, could have been a leading contender for the Democratic nomination in 2004 if it hadn't been for the plane crash that killed him.
Ted Strickland: Former Governor of Ohio with a doctorate in psychology, was mentioned as a possible Vice-Presidential candidate in 2008
Ben Sasse: Was President of Midland University in Nebraska before becoming a Senator, occasionally discussed as a primary challenger to Trump in 2020, and if he'd decided to run for Senate in 2012 instead of 2014 he could have been a presidential candidate in 2016.
Eric Greitens: Rhodes Scholar and PhD from Oxford who was an adjunct professor at Washington University in St. Louis before being elected Governor of Missouri and then having his career flame out in a sex scandal. Absent the sex scandal, he would probably be seen as a likely future presidential contender.