The Doctor's TARDIS: an interesting bi-generational issue in Doctor Who
Integral to the Doctor Who universe as the titular character herself, the TARDIS finds itself in an interesting dilemma with the bi-generation idea introduced. Although showrunner Russell T Davies's interpretation of the bi-generation process in "The Giggle" gives the show's canon an intriguing layer and hints a Doctor Who multiverse, it also presents an inevitable problem for the venerable blue box.
Many Doctors Without a TARDIS
Davies's account suggests that, thanks to the bi-generation event, many Doctors have existed over time as distinct entities. But the episode itself draws attention to a crucial weakness in this situation: not enough TARDISes for these newly created Doctors. This is a rare and remarkable event even if the Fourteenth and Fifteenth Doctors are able to get two TARDISes through the Toymaker's power. The other Doctors are left without a way across the Time Vortex since they most certainly do not come across such strong enemies.
One possible answer is shared ownership or squabbles.
The lack of TARDISes for the other bi-generated Doctors begs a reasonable theory: these Doctors might have turned to shared ownership of the TARDIS, squabbles, or compromises. The program has already looked at cases of doctors confronting their past regeneration and hinted at a possible fix. Given the complexity of temporal paradoxes and the necessity of personal journeys, the likelihood of several Doctors sharing one TARDIS is low even if the legitimacy of claiming ownership over the TARDIS is still debatable.
Clarifying the Missing Bi-Generated Physicians
Lack of TARDISes for the great majority of bi-generated Doctors offers a convincing theory for their absence from the show's story. These Doctors may have been stranded in different points in time and space with limited mobility, thus their appearance in the main narrative seems unlikely. The lack of these characters, particularly in view of the long history of the show, supports this theory by providing a different viewpoint on the bi-generation phenomena.
Grounded Storytelling Precedent: Doctor Who
Though the TARDIS is associated with Doctor Who, the show has deftly explored stories in which the Doctor is limited to particular places. Especially, the Third Doctor's exile to Earth confirmed his link to Earth and resulted in gripping narratives even though he couldn fix his TARDIS. Likewise, the Eleventh Doctor's protracted stay on Trenzalore proved the show's ability for gripping narrative even without the TARDIS's continual presence.
The Fourteenth Doctor: Notable Outlier
Unlike his bi-generated counterparts, the Fourteenth Doctor owns his own TARDIS, thus he is a rare exception to the rule. This notable difference places him in a position as a possible successor should the Fifteenth Doctor pass suddenly. Without a TARDIS, the other bi-generated Doctors struggle to negotiate the Time Vortex and take part in the continuous adventures. With his easily available TARDIS, the Fourteenth Doctor becomes the most sensible and qualified candidate to assume the Doctor in such a situation.