Yea. But in the field, producers and designers are much more useful as well. Producers are needed for large teams (not small). Larger the team, better the producer needs to be. Small teams don't really need a producer since they can communicate amongst themselves fine. But on a 100+ person team, a producer is needed so everything goes smoothly. Designers are only needed when the game demands it. Smaller games, it's easy to have a programmer/qa/designer mix of a person. But those three get separated as the game becomes larger and more complex. But everyone on the team needs to know at least a general understanding of what the others do. So they know what's going on and how to communicate.
That's actually a really good point that I didn't think of. I'm imagining how huge the development team for Grand Theft Auto V is -- pretty sure the credits are half an hour long. Everyone needs to have that collective vision which is delegated by the producer. Otherwise concepts won't be cohesive.
That's what the producers and designers are for :P. Designers usually work in a team to come up with a comprehensive gameplay, missions, etc. Then they pass those off to the producer who makes sure everyone is working correctly and efficiently. Larger games are insane when it comes to that. Which is why they are so needed. But in, say, a team of 5 people, it's easy to just unanimously agree what needs to be done.