In Doctor Who season 13, episode 2, the Doctor mentioned having been president of Gallifrey. The current Doctor Who: Flux mini-season makes frequent reference to the series' dense past and lore. Though the Chris Chibnall and Jodie Whittaker era of the show has been marked significantly by the Timeless Child retcon, these allusions connect with those who have watched since classic Doctor Who and demonstrate that the show isn't completely disconnected from its roots.
On Gallifrey, the president was in charge of the Supreme Council of Time Lords. The position was either appointed or, in an emergency situation, elected. The Doctor's election to the position occurred during Tom Baker's run as the Fourth Doctor in 1978. The Doctor nominated himself for the position in the classic Doctor Who season 14 serial "The Deadly Assassin" and returned to Gallifrey to be inaugurated during the season 15 serial " The Invasion of Time."
There was a new president in Doctor Who season 20 serial "Arc of Infinity," the Doctor's former teacher, Borusa (Leonard Sachs). Why the Doctor was no longer president at that point wasn't explained, and it's possible that his term of office was simply up. In the 1983 special "The Five Doctors", the Fifth Doctor (Peter Davison) again became president, succeeding Borusa (then portrayed by Philip Latham). This time, the Doctor's time in office ended with him being deposed. In Doctor Who season 23, "The Trial of the Time Lord," the Sixth Doctor (Colin Baker) was informed by the Inquisitor that he had been removed from the position because he "willfully neglected the responsibility of the office."
Presumably, the Doctor's removal from office was because, instead of staying on Gallifrey and overseeing it, he was traveling around in the TARDIS. It can be hard to track the timeline of Doctor Who's Gallifrey and the Time Lords. It seems safe to assume that he didn't return to Gallifrey at any point after his appointment, otherwise he would have found out that he lost his position. His absence left the High Council on their own and gave others the time to move against him.
Despite being a prominent Time Lord, the Doctor was perhaps not the best candidate for the office. Each iteration of the Doctor has cared deeply for their home and their people, but they also wouldn't be The Doctor if they weren't traveling away from Gallifrey. This is most likely why the Doctor turned the position down when it was offered again at the end of "The Trial of the Time Lord." With Gallifrey's destruction, the whole issue of the presidency is irrelevant in Flux, though perhaps Doctor Who season 13 will address it again in the future.
Doctor Who: Flux releases new episodes Sundays on BBC and BBC America.