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Probably one of the most famous examples, but the robots in The Matrix originally kept humans around as wetware CPUs using their spare brainpower. Studio execs forced the Wachowskis to change it to them using humans as batteries, even though that makes no sense. Agent Smith possessing someone in the real world in the sequels would have made a ton more sense with the original explanation.
That doesn't really work either. Human brains are not great at computing unless you are looking for "good enough," results, and only on some pretty narrow fields, facial/speech recognition, some physics interactions, etc. But worse than that... we're kind of using them. If they wanted us to compute, the whole function of the Matrix is just taking up run cycles. And you can't just coopt them during sleep, we need the rest periods ,or we literally die. Only one answer makes sense to me, it's a nature preserve. They didn't want to be responsible for destroying their creators, and the only other sapient species known to exist. So they build the Matrix to keep us docile. Then, the energy reclamation actually makes some sense. They're never going to be net positive, but assuming they are having difficulty keeping their society powered, they would be incentivesed to reclaim every watt of power they could from us to reduce our burden on their grid.
Human brains are excellent at computing certain things that are almost impossible for a regular computer. Having worked for years on computer vision I can tell you how hard it is to make computers realize simple stuff, heck, you need massive server farms just to do a basic object recognition that any 3 years old can already do. Sure, you can train a simple AI to recognize some objects, but it will never (currently) be as many objects or as precise as a person can instantly recognize.
The truth is human brains are excellent at what they evolved to do, i.e. pattern recognition. So much so that when trying to figure out data it's usually easier to plot the data in many different ways to see if something shows up. In fact usually when you try to do cluster analysis the first machine result is, let's say not great, but you can see that things are wrong and adjust the parameters.
As for your other point your brain does this automatically, they can just put a billboard with the thing they want analyzed and your brain (and millions of others) will give them the answer. Or they could use our dreams, even during sleep our brains are still active, and they could run any scenarios then. There are many other ideas, e.g. people playing videogames inside the matrix are actually controlling robots, or people working in forklifts are actually piloting construction robots in the real world, etc.
The original CPU idea was excellent, but computers weren't so ubiquitous back then, and the producers thought that the audience wouldn't understand it.
Humans solving CAPTCHA for the machines