Well sure. It’s called buying into the premise. You are already watching the show because you are willing to accept a time traveling alien outwitting aliens throughout time. But part of the premise is that when he visits Earth in the past it is in fact our Earth and our past. If there is something anachronistic not due to part of the plot it works against the premise and stretches the suspension of disbelief. It’s easier to accept a Dalek in Elizabethan London than it would be to see Shakespeare wearing a wristwatch.
Well, it’s possible Shakespeare could have worn a wristwatch. “The concept of the wristwatch goes back to the production of the very earliest watches in the 16th century. Elizabeth I of England received a wristwatch from Robert Dudley in 1571, described as an arm watch” Except they were marketed at women during his lifetime.
Umm, all I saw in your post was wristwatch (nothing about it being a digital one). Unfortunately I am not a Capaldi fan so usually binge watch 3-4 episodes at a time just to keep up and I haven’t gotten to the last 2 EPs yet.
But I have gotten used to seeing anachronisms on Doctor Who and shrugging them off. Tank and electric guitar in the Middle Ages…