The Doctor archetype in Doctor Who is (or ought to be), as I have previously argued, a Merlin type wizard.
This fits the basic plot of a chap who is not a warrior nor a ruler; but does a behind-the-scenes job advising and assisting a variety of King Arthur type people, dotted around time and the galaxy - and fighting various monsters and supernatural foes using (techno-) magic, intelligence, lateral thinking, creativity and humour.
The weakness of Doctor Who has always been the feeble moral basis for The Doctor doing this - the lack of any answer to the question of - on what authority The Doctor is acting? And why should his moral evaluations be regarded as correct as, or better than, the evaluations of the enemies he fights?
The only given answer is a kind of (supposedly) common-sense ethic of the Doctor taking sides to protect the weak and non/less-violent, to prevent (more) suffering, to free (suffering) slaves etc; combined with a utilitarian ethic of doing this for large numbers of people (which is why he has to, week after week, keep on 'Saving The World' - or even The Universe).
But these motives don't bear much examination, especially when the sufferings of 'the baddies' are put into the equation (and who says they are baddies anyway?)
So, to improve Doctor Who it would be necessary to provide a solid cosmic metaphysical structure and narrative within-which Good and Evil are real and objective, and within which the Doctor can operate in an ethically objective way: so that he really is on the side of Good.
A Mormon Merlin would be a good basis for this - so long as the Mormon framework was not made explicit, since that would scare the horses!
But Mormonism is already a cosmic and Who-like religion, at least in some of its non-canonical folk aspects - with worlds upon worlds 'rolling into existence' and recapitulating analogous processes of salvation and spiritual progression.
Furthermore, Mormonism has a view of Time and Space which is compatible with that of the type of modern theoretical physics most used in SciFi - with General Relativity for example.
(By contrast, mainstream Christianity often has a non-dynamic God's-eye-view vision of Time as static and universal - with God seeing everything past, present and future simultaneously: a model not much use for any kind of Time adventure.)
Doctor Who could take its place in this scheme, with The Doctor as (implicitly) an exalted human angel (like Moroni),
or one of the 'immortal' translated men such as the Apostle John or one of the Three Wandering Nephites
Whether implicitly-Mormon or (more likely!) not; Doctor Who would surely benefit from a solid, objective theistic religious morality; to make The Doctor a really-Good person; not just a self-styled-goody.