I am Emma. I am big fan of pokemon creatiures. My interpretation has always been that in the Pokemon world, raising 6 pokemon is extremely difficult. The only reason the protagonist can do it is because he's a prodigy. It's why everyone is always marveling over how strong he is. Most people would have a really hard time raising that many pokemon.
It's not as easy as the game makes it seem. You've got to buy the right food, you've got to exercise them, and you've got to become an expert on all the intricacies of how to properly care for each particular pokemon.
This also explains why so many people specialize in a certain type. Because it's so difficult, most are only able to do it by becoming experts on a particular kind of pokemon. If they attempted to raise many varied types, most would be overwhelmed and do a poor job.
The idea of having a team of 6 pokemon with different types is so insanely difficult that only natural geniuses like the protagonists are able to do it. If everyone could do it, then having single type gyms would be really stupid.