Should Vera Custer start the game with an ability? Her character card says:
"For one whole round, she gains the same ability of another character in play of her choice."
The rules further clarify this by saying: "at the beginning of her turn, before drawing any cards (in phase 1), she chooses any other character still in play. Until her next turn, she has the same ability as that character."
This makes it sound like she DOESN'T start the game with an ability, but that seems a bit severe to me, especially since she only has 3 hit points.
Your reasoning seems sound, and I think you're right. If it were supposed to happen at the start of the round, it would have said "at the beginning of the Sheriff's turn". It is possible that that is what it should have said, though.