There's no choice for both! I totally enjoy playing for fun, but it's also a lot of fun to breed and get the right stats for your Pokemon! So I'm a breeder and I love it! And it's really not that hard either to IV/EV train
I know a lot of people like to just play for fun and I totally respect that, but I for one can't stand those stinking cheater NPCs kicking my butt in the Battle Maison--so I go all out in breeding and in training these days.
It's not actually as complicated as it sounds at first, but it is definitely time consuming so it makes sense that people don't really give it a try.
I have to tell you though, it feels sooo good to do well with a Pokémon you bred and trained to (near) perfection on your own!