I don't think the producer has to do much, outside of maybe knowing a writer
What kind of jobs can you get with a video production degree?
Video Production jobs are usually broken down into the phases of the process. You have certain jobs in pre-production, production, and post production. There are a few jobs that crossover into each phase of the process and very few jobs oversee the whole process.
Let us start with the jobs that oversee the entire process. Really this comes down to the Producer job and the Director job. There can be multiple producers and/or directors on a project, but these two positions are really the two that oversee all phases of the process on a project.
Pre-production entails Location Scouts, people who search for locations to film at. Casting Directors are people in charge of finding actors or people to fill certain acting spots on the project. Storyboard artist are who draws and paints visual frames that the director wants to help visualize the project. You need someone to write the script of the project, so a Writer is very important in the pre-production phase of the video production process.
Production has an enormous list of roles and jobs that are needed to be filled in order to pull off a good video. You need a Director of Photography, which is the one who is in charge of the look of the video. Someone who creates the best looking angles for the camera to be in. A Gaffer is someone who handles the lights on a video production set and an Audio Technician is the one who is in charge of the audio on the set. You have a Set Designer who creates the actual sets where the actors will be performing in. A Camera Assistant is the person who physically uses the camera and gets the shots the Director of Photography tells them to get. The job position that helps the set stay in process is the Production Assistant. The Production Assistant gets supplies, food, and anything else the Director/Producer asks for to help the production phase of the process run smoothly.
Post production comes down to the Composer (if needed) and Editor. A Composer is someone who creates music for your video and the Editor is the one who cuts what was shot and puts it together to create the story of the video. Of course the Director/Producer oversees what is being done. Now if it is a large scale project, you will have multiple people handing the post production phase to color grade, add effects, create sound effects, etc. It all depends on how big your project is.