An analogous relationship should help you answer the question: Who runs the piano in a violin show-piece (e.g., Kreisler)?
Mike
Mike's answer here points in the right direction.
I am a conductor. To respond directly to the OP's actual question, "Who runs the orchestra?", the answer is that the conductor directs the orchestra with his conducting, whether or not there is a soloist. When there is a soloist, whether voice, violin, piano, trumpet, or any other musical performer, the conductor directs the accompanying ensemble. Period.
When there is no soloist, the conductor is responsible for the finished piece of performance art. When there is a soloist, the conductor has to be made aware of the soloist's vision of the piece, of how the soloist wants to translate the notes on the page into a lived art experience. Mention has been made of this in previous posts. But that conversation is between the soloist and the conductor.
When as the conductor, I am accompanying a soloist, I do so as the person under whose direction the orchestra is playing. The soloist is not that person. For the performance to be carried off as it should, there are elements in the music sometimes of dialogue, sometimes of duet, and more often than not, of the accompaniment accompanying and supporting the soloist. For these aspects of the music to be translated into artistic performance, both soloist and conductor have to respect the requirements of the music, and as the music requires, initiate their own dialogue/duet/accompaniment in their preparatory meetings. Is it possible for disagreements to arise? — Certainly! But that is the challenge to be faced (and not feared) in each individual situation.
To be avoided by both conductor and soloist is letting members of the accompanying ensemble make their own 'artistic' decision(s) based on what they hear themselves. The focal point for the musicians in the orchestra must be the direction given by the conductor, not their own individual relationship with the soloist based on what they hear in the soloist's performance. Thus, the relationship between the soloist and the conductor is crucial, and it is no secret that the ego of either or of both can get in the way of the music and of the performance.
As an orchestral musician, I have played under many different conductors, and enjoyed or suffered through interpretations that were or were not as I thought they should be. Life gets really interesting when conducting a piece that the ensemble already knows and has performed many times before. When I am on the podium, my interpretation has to prevail, regardless of who did it which way before, and that's just the way it has to be. If my interpretation is different from what the musicians expect or want, they have to adjust because my responsibility is to direct them so as to rehearse and perform my sense or vision of the piece.
To my mind, this same rule has to apply between the soloist and the conductor, such that the accompanist/conductor should defer to the soloist's presentation of the music, unless of course something quite outlandish is under consideration.