BackFire
Blood. It's nature's lube
Yeah, the first time I play through them I tend to do everything I can, just because I want to experience everything, as a result my characters generally get quite strong just from the weapons and armor and whatever "uber" spells you pick up along the way. This creates the problem I spoke of earlier, trivializing the end boss by killing it in two hits. The random encounters do allow people to just grind XP until they get to level cap, and simply blaze through the whole game without any challenge. While it's nice to be able to level up easily when you need to just by redundantly killing monsters over and over, it takes away from the story.
I'd also say that by doing everything in the game, you often lose the cohesion of the story. For instance, in FF7 by the third disc there is a massive meteor heading toward the planet, so logically time would be of the essence, coincidentally this is around the same point in time when you start to do the sidequests and bonus games. So people get to this point where the game tells them that a big ass meteor is supposed to be colliding with the planet in a matter of days, and at the same time it promotes you going and screwing around at an amusement park and breeding chocobos. If you do all of the sidequests at this point in the game(as most people do), you begin to lose sight the coherency and importance of your journey and begin only caring about getting the best gear you can. By the time you get all of the items you can get, and get back into the main story, you already haven't done anything to progress the story in so long that you just don't really care anymore. The game has unintentionally taken you OUT of the story by promoting the sidequests all at the same time.
This happens in all of the FF games in some way. So generally I like to beat the game the first time doing everything, and then later going back and playing it for a second time just for the storyline and skipping all of the extra features. That way you get the best of both worlds.