In-universe, I'd say no. Out-of-universe, DW is a British show, and playing the Doctor is a plum achievement for any British actor. And when we think of British acting, we think of the theatre, and when we think of British theatre, we think the Globe, in London, half a millennium ago. Britain is NOT America, where one can be "discovered" buying ice cream at the mall with zero stage experience, and be on the cover of Tiger Beat the following week. So, you have an iconic television show with an iconic character marketed to an audience, the ancestors of whom were the groundlings at the Globe Theatre.
The producers want an actor of a certain age, with the chops, who wants to take the show in the direction in which they want to go. This actor has to be free of other commitments, in order to dedicate his time to the character. Once you look at these requirements, in BRITAIN, you will find a very small pool of candidates who are any sort of minority, the majority of dwellers of the British Isles interested in participating in British culture being white. They aren't going to find some good-looking minority lad or lass at the local chippy and offer him/her the role of Doctor Who. There could have been minority actors in the running, but they may not have been able to make the commitment, or may be the wrong age.