Summary
Despite the War Doctor's redemption, Doctor Who rendered their name meaningless due to the supposed genocide caused during the Time War.
The War Doctor cannot be held solely responsible for the conclusion of the Time War, as they did not actually commit dual genocide. Instead, their sacrifice and renunciation of the Doctor's title serve as a poignant representation of their readiness to carry out abhorrent deeds in the name of a greater cause, serving as a haunting reminder of a bleak era.
Doctor Who has made the War Doctor's name meaningless despite their redemption at the end of the Time War. The War Doctor was introduced in the 50th Anniversary special of Doctor Who, where the unanswered questions surrounding the events of the Time War were finally addressed in "The Day of the Doctor". Technically, the War Doctor is the 9th Doctor, but his untold story causes Christopher Eccleston's regeneration in New-Who to be recognized as the 9th Doctor. Throughout "The Day of the Doctor," David Tennant and Matt Smith's 10th and 11th regenerations carried a deep sense of shame from their time as the War Doctor, doing their best to conceal their past.
As a result of the purported genocide the War Doctor committed during the Time War, they renounced the name of the Doctor, feeling unworthy to bear it. Revoking the title of the Doctor meant that Eccleston's incarnation became the 9th Doctor, while the War Doctor remained nameless and was referred to as the War Doctor, Doctor of War, the Warrior, and the Renegade. The Doctor may feel ashamed of this version of themselves, but the War Doctor's lack of a title is not solely due to this reason, and it may not be entirely fair to reject the War Doctor in this manner.
The War Doctor Deserves A Proper Title
The intentions of the Ward Doctor during the Time War may not deserve the Doctor's shame. In "The Day of the Doctor," it was revealed that the War Doctor did not commit double genocide as previously believed. The War Doctor, along with the Tenth and Eleventh Doctors, found a peaceful solution to the Time War. Although the War Doctor was willing to take on the Doctor's name again, the out-of-sync timelines prevented this from happening. Reclaiming the Doctor's name also had no impact on the War Doctor's regenerations, and they are still not considered the ninth Doctor.
Furthermore, not only did the War Doctor not engage in the alleged actions, but other regenerations have undertaken morally worse deeds. While the Doctor strives to be the hero, subsequent incarnations have committed acts far more heinous than the War Doctor. For example, the Thirteenth Doctor executed a devastating triple genocide in Flux by eliminating the Daleks, Cybermen, and Sontarans all at once. What sets this apart is that the Thirteenth Doctor exhibited no remorse for their actions, whereas the War Doctor grappled immensely with the moral weight of their potential actions. In light of these events, Flux raises doubts about why the Thirteenth Doctor is considered more deserving of the Doctor name than the War Doctor.
What The War Doctor Having No Title REALLY Means
However, the War Doctor's reluctance to adopt the name of the Doctor is a complex issue. The disrupted timelines led the 9th Doctor to believe that they were responsible for the end of the Time War and the extermination of their own race, which weighed heavily on them. The War Doctor is no worse than other incarnations of the Doctor and arguably better than some, as someone had to take responsibility for ending the Time War. The War Doctor did not actually commit the accused acts, but they became an easy scapegoat when the Time Lords and the Doctor themselves needed someone to blame.
By accepting the blame and relinquishing the title of the Doctor, the War Doctor demonstrates their willingness to make a sacrifice. Their distinct identity also facilitates the growth of the Doctor, transforming them into a different person, almost a new Doctor. The War Doctor's actions continue to haunt future incarnations, as the Doctor knows they were capable of committing such a reprehensible act as genocide in Doctor Who, making the War Doctor a symbol of an era the Doctor would prefer to forget.