Tim Harford cites this study in the podcast episode notes:
https://www.sciencedirect.com/science/a ... via%3Dihub
The abstract states that:
Candidates that were taller than their opponents received more popular votes, although they were not significantly more likely to win the actual election. Taller presidents were also more likely to be reelected. In addition, presidents were, on average, much taller than men from the same birth cohort.
The popular vote issue seems important, as that means that Gore beat Bush.
Also, the article was published in 2012, so presumably they didn't have data on Obama versus Romney (November 2012).
Looking at the above mentioned Wiki page, it looks like the height advantage is a 20th century thing. That could be a consequence of mass visual media - cinema newsreels and then TV - which made the candidate's visual appearance more important than in the 19th Century era of newsprint.