The Ever-Looming Plot
Let's be honest here. The plots in Pokémon games have never been above the second-grade reading level. They exist solely to give your character a reason to travel around the continent. Oh no, Team Magma is trying to cover the world in lava! No one cares. People are playing the game because they want to complete their quest of catching all the Pokémon, and defeating all the trainers. With that said, though, I won't suggest the complete and utter removal of all plot-esque elements. What I'd prefer is a more dynamic and optional inclusion of plot-related events.
The main plot of the game, plain and simple, would the chronicle of your journey to become the Pokémon master. Everyone would have the same main plot elements, namely each gym and the final clash at the championship. All of the pieces in between the gyms, however, would depend entirely upon where you happen to be and what you happen to be doing. A dozen events may be occurring in towns and fields across the continent, but you only become involved in those you happen to be near and decide to investigate.
These events would be the primary venue for using Pokémon outside of battles. Perhaps a little boy runs up to you, and tells you that his Skitty is trapped up in a tree. Whip out your Bulbasaur, and save the poor thing with its vine whip. The little boy may give you a reward. There would be far more in-depth events than the preceding example, however. Let's say someone's Charmeleon got ticked off and set a building on fire. Your mission, if you choose to accept it, is to rescue everyone in the burning building. In all likelihood, you won't be the only trainer around. So it's up to you and whoever else decides to jump in to save the day. By using various Pokémon and their abilities, you traverse the building and rescue people one by one. Maybe someone has a Pidgeotto, and fly people out of the building to safety. However, they don't have any water Pokémon that can put out the fire that blocks their way. By teaming up and forming temporary parties, groups of people can complete events together.
However, we can't have fifty trainers rushing this building all at once. There would have to be conditions built into the system that would convince certain trainers to pass on certain events. Rushing into a burning building is a very dangerous gambit, after all. Especially if you're a low-level grass Pokémon trainer. If you're in a disadvantageous situation and you still manage to pull it off, however, your rewards would be that much greater. Maybe your grass Pokémon becomes resistant to fire after this event, giving it an enormous advantage in battles from thereon out. Maybe you decided to use a fire Pokémon, and after interacting with the flames in the building directly, it learns a new flame-manipulation technique that can be used in battle.
At the end of the season, you can look back on your achievements and find that your own story was completely different from everyone who was playing around you. Continuing threads could begin as you complete events that you come upon, and events could start to occur in progression. Maybe because you stopped a fire in this one town, a criminal managed to survive and is now wreaking havoc in another town. You might become famous because of several noble deeds, to the point where an event occurs in one town that can only be completed if you happen to show up. A system like this would work even better if Nintendo employed script writers throughout the entire run of the game who would create new events on the fly based upon how previous events in the world were treated.
Conclusion
After writing this piece, I find myself being more depressed about the future of the Pokémon franchise than anything. The knowledge that Nintendo will likely never implement changes as radical as the preceding has probably ruined me for future iterations of the franchise that are made from the current mold. After all, every fully-fledged Pokémon game that has been released in the last six years, regardless of how cookie-cutter they all were, has sold in the kabillions. It seems that the buying public is perfectly happy with the current state of things.
If the Pokémon franchise ever does go online, it likely won't be in a MMORPG fashion. The game will play the same way as it always has, and you'll probably be able to join in mini-tournaments and play against other interested trainers like an online version of Pokémon Stadium. It'll be neat, I guess, but nowhere near as incredible as it could have been.
Pokémon is Nintendo's cash ticket with the younger crowd, and upping the sophistication of the game to attract more of the older market probably isn't on the top of Nintendo's to-do list. Though, Pokémon Colosseum was considerably darker than the GBA games from a thematic standpoint. Maybe, just maybe, Nintendo's looking to expand the general appeal of the franchise. I highly doubt the changes I've proposed will ever be considered, but I'd be more than happy to be proven wrong.