He wasn't a politician though. He was working for Ford when Kennedy asked him to help out wasn't he?
I guess potentially if Kennedy / Johnson had had a much sourer relationship he may have been asked to be veep, but this would have lost him the election.
Basically you'd need him to go into politics proper following his positions with Kennedy and Johnson and not have such an association with Vietnam (somehow). Perhaps he runs for senate following Kennedy's assassination. A possibility as I don't think Johnson liked him very much.
He'd probably be quite a competent president, but he's never going to get there without some major changes.
Also, to the comment about personality, he seems quite pleasant in that documentary he was in (where the annoying interviewer keeps shouting things at him).