In the theater, both live stage and movies, there is a role called "producer". In many cases, the producer is the financial backer of the movie or play, and may be silent having no further interaction with the project other than signing checks. Most producers are actively involved with the making of the movie or staging of the play. The primary goal of the producer is to oversee and deliver a project while preserving the integrity and vision of the film or stage production.
The business analyst does not usually finance a project. However, the business analyst is responsible for the integrity of the solution and the approval of the audience, namely the business community.
The producer starts work by obtaining the rights to a work to be transferred to the screen or stage or approves a script for production. This then is the problem statement: the script is not a movie. The business analyst starts work by defining the business problem. From there the roles are fairly similar.
Wile it is the director who works with the actors, the stagehands, the cameramen, the extras, the stuntmen, the makeup and costumers and so forth to implement the production, the producer interacts with marketing and potential audiences and publicity and all those factors that influence the final result outside the sound stage. It is the project manager who performs that role with the business analyst, directing and managing the solution team of programmers, testers, data base analysts, architects and so on. The business analyst provides the feedback from the audience - the stakeholders and business community - and keeps the project aimed at the target of solving the business problem.
Perhaps in these days of agility when the business analyst who only defines requirements is considered to be superfluous, recasting ourselves as IT producers wedded to organizing the entire product from problem definition to solution in the business environment might be a valid compromise.