You're right that behavior of split-brain patients does not apply to healthy individuals with an intact corpus callosum. In the healthy brain, the hemispheres constantly communicate with each other and it’s wrong to say that one hemisphere “takes over” or “dominates.” The notion that creative people are “right-brained” and analytical people are “left-brained” is also nonsense.

However, there are subtle (and not-so-subtle) differences between each hemisphere’s cognitive abilities, and many of these differences were discovered by studying split-brain patients. As others have said, language is largely lateralized to the left hemisphere and visuospatial processing/reasoning is more right-lateralized (which probably led to the idea that artists are “right-brained”). Here’s a quote from a nature article on split-brain patients:

“The split work really showed that the two hemispheres are both very competent at most things, but provide us with two different snapshots of the world,” says Richard Ivry, director of the Institute of Cognitive and Brain Sciences at the University of California, Berkeley. The idea of dichotomous consciousness captivated the public, and was greatly exaggerated in the notion of the 'creative right brain'. But further testing with split-brain patients gave a more-nuanced picture. The brain isn't like a computer, with specific sections of hardware charged with specific tasks. It's more like a network of computers connected by very big, busy broadband cables. The connectivity between active brain regions is turning out to be just as important, if not more so, than the operation of the distinct parts. “With split-brain patients, you can see the impact of disconnecting a huge portion of that network, but without damage to any particular modules,” says Michael Miller, a psychologist at the University of California, Santa Barbara."

Besides the complete lateralization of cognitive processes, it is also possible for homologous regions of each hemisphere to possess different processing characteristics. For example, Jung-Beeman (2005) talks about how the dendritic arbors of neurons in the right hemisphere are larger than those of the left hemisphere and the right hemisphere is generally more inter-connected than the left hemisphere, which may bias the right hemisphere to coarse (holistic) processing and bias the left hemisphere toward fine processing.

I actually just co-wrote a paper with Michael Miller and Michael Gazzaniga that argued that cognitive modules in the left hemisphere strive to reduce uncertainty (by creating explanations and making inferences) and modules in right hemisphere strive to resolve conflict between inferences and reality. Our framework is based off of studies on split-brain patients, as well as patients with brain damage, delusional patients, and healthy individuals. We don’t suggest that only the left hemisphere creates explanations or only the right hemisphere monitors explanations. Instead, we suggest that each hemisphere can reason, but the conclusions of each hemisphere may (but may not) be different because they have different reasoning biases and strategies.

I think the idea that

“different structures within the brain come to different conclusions and apparently fight it out to determine which one will dominate”

is plausible. In the paper, we suggested that the hemispheres' different reasoning strategies can be preferentially biased depending on reasoning demands (such as whether a given situation requires creativity or caution), and it is common for certain brain regions to be inhibited or facilitated depending on task demands. However, I imagine that there usually isn't much conflict between the left and right hemispheres. Even if there are different processing strategies in the brain, I think the outputs are well integrated in the (healthy) human brain.