Thanks, F&SF.
But, to take that argument to the extreme, would it mean that I would also have to learn about make up, sfx, cgi, cinematography, etc? I don't think I should have to learn all the skills.
Never good to take things to the extreme but if you watch enough "behind the scene" clips, you get a good sense of what's involved. However, you need to focus on the "bread and butter" parts of the producer role. Many times the producer needs to secure and oversee financing. There are producers by title only, but actual working movie moguls need to be thinking in terms of profitability and budget. You should have a good handle on the cost of SFX, CGI, etc. though not necessarily how it's done. For example, wardrobe needs to be coordinate if you are doing green screen/blue screen so you don't have conflicts. How much will need to be green screened and that will figure into studio rental. While normal makeup is inexpensive, if you need special prosthetic, the cost jumps up rapidly because you are also paying actors for modeling and fitting times. While you don't need to be a technical expert with cameras, understanding what equipment rentals will be needed should be part of your education.
Again, for a producer, your principal concern is the bottom line. And in the media industry, it's kind of like horse racing. A mogul will have a few projects and is a booky of sorts. S/He take other people's money and put odds on the horses (scripts). There's no magic solution but understanding good scripts from bad ones, betters your odds of recovering some of your investment at the races (film festivals).
So, do you need to take a class in automotive repair to work on your car? No. Can it make you more proficient and marketable if you choose to go in that direction? Of course. Learning to change your oil, refill fluids and basic maintenance doesn't require you to be an expert but makes you good at keeping up your own car. Taking a basic screenwriting class won't make you an expert but it will give you a good sense of what makes for good formatting, structure and story. It can help with your own scripts or make it easier in evaluating scripts.
If you invest in a franchise, it's best to get trained in how to run that business. If you're goal is to be a film mogul, it's worth investing some time and a little cash into some training to learn a bit about the industry. This is a competitive field and a little education can shorten the learning curve. For every good script, there 100 more that are mediocre or worse.