A warrant is not needed to search a vehicle...and its not private. One can be intoxicated in public in a vehicle...
A warrant is required to search a vehicle unless there is probable cause.
A warrent, or probable cause, IS required to search your vehicle, and it is private property. Your example provides probable cause, but does not make your vehicle public property.
Just say no to ALL requests to search your car. When a policeman has stopped you, he is not there to be your friend, he is there to put you in jail if he can. It is foolish to answer leading questions or submit to a search voluntarily.