"...it occurs to me to wonder whether Blair got into the position he's in partly because people perceived him as likeable? "

I'd say the answer is yes - but the people concerned were the Labour Party activists, and they were judging what they thought ordinary people liked. I doubt very much whether the people who made him leader of the Labour Party particularly liked him then or like him now.

I also doubt whether that many ordinary people particularly liked him when they voted Labour to get rid of the Tories and to keep them out. I don't think "liking" really comes into it too much.

The symbolic question people ask about politicians often is "would you buy a used car from him", which is about pereceived hmnesty and also to some extent about competence, but not about likeability.