This makes absolutely no sense, but I did it off a Midjourney binge, so.
The First Doctor, acted by John F. Kennedy (1963)
The choice of a youthful American actor to be the main character of a new British show would be a surprise, but 'JFK' or 'Jack' as his fellow cast would know him as would be very keen on the TV show, highlighting it as a way to "inspire men to explore space". This attitude would imbibe the show with a certain sci-fi utopianism, something that got some old directors resentful. JFK's sudden death forced the show to come up with a quick explanation for a new face the following season. JFK is seen within Doctor Who circles as the 'best Doctor we basically never had'.
The Second Doctor, acted by Pierre Trudeau (1964-1969)
Unlike Kennedy's vaguely youthful but 'respectable' figure, his successor - a fellow North American actor - would turn the role into a bit of a 'renegade', and it is here that we first learn that he is on the run from his fellow 'Time Lords', and it is here that we see the trend start - although it is noted that it was planned for Kennedy as well - of the Doctor having attractive female companions who he acts vaguely attracted to. Trudeau would leave after a contract dispute, but agree to a final payout in exchange for doing the regeneration scene.
The Third Doctor, acted by Éamon de Valera (1970-1973)
In a move away from the 'youthful' faces of Kennedy and Trudeau, the directors picked an older Irish actor. Éamon de Valera often spoke of how he wanted to make the Doctor into a 'respectable fatherly figure', even as he was restrained by the directors from speaking on his Catholic beliefs overtly. Befitting 'Dev''s scepticism of sci-fi shows, the Third Doctor would be primarily stuck in modern times working for EIT [European Intelligence Taskforce] or exploring historical settings. Citing ill-health, De Valera would step down in 1973.
The Fourth Doctor, acted by François Mitterrand (1974-1982)
The Doctor to a generation of fans, Mitterrand's Fourth Doctor would be regarded as a 'kaleidoscope', being radical and somewhat avant-garde earlier on in a correction to concerns that the Third was too staid, he would transform himself into a respectable figure, although with a tendency to give condemnatory speeches against injustice he saw in the universe. This is why in most spoofs of Doctor Who, the Doctor has a French accent. He would eventually decide it's 'time' for a fresh face to be the Doctor, and step down in 1982, doing a final regeneration scene.
The Fifth Doctor, acted by Deng Xiaoping (1983-1985)
The 'Chinese Doctor' as he's often unfairly remembered as, Deng was a notable actor who made it on the scene in Hong Kong, and in a peculiar attempt to appeal to the growing Chinese consumer audience of the Republic of China [slowly opening up under its first non-KMT government], Deng was headhunted to succeed Mitterrand. He accepted only if he could have full control over how his Doctor is characterised. The Fifth Doctor is notably absent his predecessor's morals, being a pragmatist in how he tackles the universe. "The universe doesn't care if a cat is white or black. Only that it catches mice." is an often-quoted statement. After two years, he got in a contract dispute and chose to leave the show. It is noted that under Deng, there were more episodes set in Asia than before, and it wouldn't get like this again until Moon Jae-in.
The Sixth Doctor, acted by Elvis Presley (1985-1987)
American singer-actor Elvis Presley long had an adoration of Doctor Who, especially with its first and third Doctors. After Deng stepped down to middling ratings, Presley auditioned for the role and got it by directors hoping that another American would stir up interest in the all-important US audience [a white whale, really]. Presley's Doctor would be notably that of an older man seeking to capture lost glories, with the Doctor's age often underlined, even as he tried to be the charmer with the ladies.
It is often mocked that the Sixth Doctor's greatest success at charming ladies was with the respectable gentleladies and that for all Presley's old rock-and-roll past, his Doctor was at home having a cup of tea. Rapidly getting tired of the role, Presley announced he would step down in early 1987.
The Seventh Doctor, acted by David Bowie (1987-1989, 1991, 1996)
The last Doctor before the show was cancelled in 1989 was perhaps its most eccentric and dark yet. The British singer David Bowie was a name often floated for the role, and after one letter was sent at a very opportune time, Bowie finally agreed.
The Seventh Doctor is noted to be one embracing bright, eye-searing colours, the darkest stories yet [including the notorious 'Time Conclave'] and for stuff that caused the moralists to scream their throats hoarse about. In the end, after much pressure, the BBC finally cancelled the show. This is widely considered [at least by Who fans] as part of why Michael Heseltine's Tories lost the 1990 election to Bryan Gould's Labour. Bowie would return as part of a TV movie in 1991 and a sequel in 1996 where he would regenerate.
The Eighth Doctor, acted by Bill Clinton (1996, 2007)
The 'Jazz Doctor' as he's affectionately called, this Doctor was more overtly American in tone than previous American-acted Doctors due to the fact he was first introduced in a movie made in a collaboration with NBC and NBC insisted on the character being more 'Americanised'. A 'slick' man with none of his predecessor's 'eccentric' nature, his playing the saxophone was at the actor's insistence [the actor was a former saxophonist]. The claim to being 'half-human' would emerge here much to fans' discontent.
The movie would not get a TV show due to Clinton's sexual indiscretions in his personal life, but by 2007 he was asked back to do the first half of the rebooted TV show, regenerating halfway [in what is often called the "most pointless regeneration ever"]. In between he often engaged with the Doctor Who community, and even now, many fans still defend his personal behaviour because of that personal interaction.
The Ninth Doctor, acted by Julia Gillard (2007-2010)
Anxious about the Doctor's 'reputation' after Bill Clinton, the show elected to take a new tack and choose a woman to be the main face of the rebooted series. Welsh-Australian actor Julia Gillard was ultimately chosen. Gillard's Doctor was noted to be 'fiery', 'passionate', a deliberate callback to the 'classic' Doctor of Mitterrand with a return to his condemnatory speeches against raging injustice. However, Gillard's Doctor also carried with her this darkness that the Eighth Doctor acquired in the war [an obvious analogue to the Iran War started by President McCain].
The 'Time Lords' were dead, and all that was left was her and her rival the Master [now acted by fellow Australian Kevin Rudd], and the culminating scene in where Rudd's Master exposes Gillard's Doctor as the one who ended the Time War by destroying the Time Lords utterly, and declares that as long as he lives, she will see the Time Lords' vengeance on the one who killed her own people. After this, there's an [admittedly generic] fight scene, and in the end while the Master succeeds at mortally wounding the Doctor, he relents at the last moment and leaves her to regenerate.
Gillard's Doctor is noted to be one of the most morally complex Doctors yet - being someone determined to uphold morals and 'do better', even condemn the universe for its rampaging injustices, yet she has so many deaths on her hand. Gillard received an Oscar for her work in 2010 after The Very Last of the Time Lords aired, her last episode.
The Tenth Doctor, acted by George Papandreou (2010-2012)
Greek actor George Papandreou's time as the Doctor is not remembered that fondly. His defenders say that George Osborne's Tories' austerity policies hitting the BBC meant that he had to act on a shoestring, but his critics say that's no defence for poor acting. His Doctor's focus on being the 'reconciler' was widely liked at first, but quickly ran its course, with the 'expert' persona being something unpopular in a Britain slowly resenting the austerity programme. Let go after 2012, he would eventually partially redeem himself through his dogged efforts in audiobooks and it is this 'later' Tenth Doctor that would appear in Reunion of the Doctors - an episode many say is Papandreou's 'best' episode, half in jest, but half seriously.
The Eleventh Doctor, acted by Moon Jae-in (2013-2017)
The 'K-Doc' as his critics pan him as, it's undeniable that Moon Jae-in's entrance also brought with him new talent from South Korea seeking more 'global' stories and more challenging plots, something that divided the Doctor Who community, but the Doctor himself was a key facilitator in soothing tensions over. It is in the Eleventh Doctor's time that Doctor Who starts to confront racism and discrimination overtly, even as Moon was reluctant to go along with the new plots, instead seeking to 'water' down the controversy for widespread consumption. In 2017, he stepped down, citing a desire to move to new roles, but some think it's because of disagreement with the writers.
The Twelfth Doctor, acted by Stefani Germanotta (2018-2021)
With American actor Stefani Germanotta, the Doctor became this sort of 'lovable eccentric' [even more than before], but at times more eccentric than lovable. Dipping back into some old plots the Seventh left off, the Twelfth mixed eccentricity with a deeper darkness within her than any of her predecessors. The 'growing darkness' would be noted as a tendency of Doctors since the start of the rebooted series, and under Germanotta it would deepen, with her eager willingness to get along with her 'fam' being undermined by her past creeping up on her.
Taylor Swift as 'Missy' [the new Master] would be a key rival to Germanotta's Doctor, and in a cinematic scene reveal that the Doctor failed at destroying Gallifrey. Germanotta would choose to step down in 2021, citing a wish to move on.
The Thirteenth Doctor, acted by Vladimir Putin (2021-2023) [incumbent, outgoing]
It is with Putin's Doctor that we see the 'true character' of the Doctor. Beneath all the joyfulness, all the fragile eccentricity, is a scarred veteran of a war that continues to fight itself in his head. The emphasis of the Thirteenth Doctor is that he is a man at the end of his line, there's no more hiding places. The Time Lords are after him, there's nowhere to hide. Putin, being a noted Russian dramatic actor and veteran of the Afghan War, brought with him actual war experience that he channelled into his characterisation of the Doctor.
Unlike previous Doctors, the Thirteenth is unrepentant [in a way that doesn't come across as flattering] at his actions in the Time War, even if over time he concedes that perhaps he could have done it a different way. At times, he comes across as a destroying force, not as someone like a human, putting truth to the words the Cyberdaleks had for him - "the Oncoming Swarm". He is a force manifest, not someone to gently know as a friend.
The 2023 series, titled The Trial of a Time Lord is his final series, and the directors are hiding the identity of the next Doctor very firmly, even at times questioning if the Doctor will even get a fourteenth incarnation, but the firm consensus of the fan community is that he will, and the show will roll on.
After all, the ratings are the best they've ever been!