I don’t know if a producer is ever a draw for a movie, no one sees that a certain person is a producer and decides to go see a movie. But producers do a lot of shepherding a movie through all the stages, and picking out cast and crew, and have input on what the movie should be like. So if you see “from the producer of ‘X’”, it often gives somewhat of an idea of what type of movie and maybe what the quality of the movie will be.
For example, both of Oracle chairman Larry Ellison’s kids have become film producers. Megan Ellison has produced The Master, Zero Dark Thirty, Killing Them Softly, Her, American Hustle, and Foxcatcher. David Ellison has produced Mission: Impossible – Ghost Protocol, Jack Reacher, G.I. Joe: Retaliation, Star Trek Into Darkness, World War Z, and Jack Ryan: Shadow Recruit. Although both were producers on True Grit and Terminator Genisys. But in general, if you see one of their names, it gives you an idea of what type of movie it is, compared to if it’s the other sibling’s name.
Like you said, the writer isn’t more prominently credited and famous in most cases because the script is out of their hands. It doesn’t have to necessarily be other screenwriters who doctor a script, it can also just be changed a lot by the actors or director as the movie is being made. Or it could be shot with no lines changed, but be very different in quality or tone based on the director and actor choices.
I know very little about sports, but are the players employees of the coach? I thought that the coach tells them what to do, and has some say on who can play and who can’t, but that players and coaches are all employees of the team. At my job, there are people who can tell me what I need to do, and make things hell for me if I don’t follow their rules, or maybe even get me fired if they really don’t like me and don’t want me there, but they aren’t my boss with full discretion on firing me or not.
Battle Creek production was announced in September 2013, based on a script Gilligan wrote 10 years before. It got more scrutiny than it otherwise would have, but it also got made because of Gilligan’s name.