I think we are forgetting the context of the show in terms of the VP pick. In the world of the show, Elizabeth McCord would probably be one of the most high profile and popular secretary of states ever. In show world, she has dealt with Iran, Cuba, Russia, not to mention all the small crises she has handled. She was almost assassinated/killed on multiple occasions. She helped uncover the former secretary of state conspiracy plot. She was even "acting President" for a day! She would be in the news all the time and extremely popular. It makes total sense that she would be a popular choice for the VP slot, and yes, she would pop up on a lot of lists as a future presidential candidate if she had that ambition in the show world.