At what point would you say a Pokémon game's story "ends"?

after the Champion. Johto games are the only exception.

>>29946823
But that bugs me though because I wouldn't count Platinum as having been beaten until you go to Stark Mountain and stop Team Galactic for the final time. That's where the sense of finality hits me (along with Arceus, but that's an optional event). Even in Gen I, I wouldn't count it beaten until you catch Mewtwo. That's THE final challenge in the game outside of the Pokédex.

Generally the story ends with the Champion, which is why the Credits normally roll after beating it.

The whole point of most games are that you want to beat all the Gyms in order to make it to the Pokemon League and become the Champion.
The villain Teams you come across on the way are just the antagonists of the story who attempt to stop your goals indirectly through their actions. For instance, if you never helped Lance beat Team Rocket in GSC then he would have never been at the Pokemon League ready to battle you.

Post game battles/events are an incentive and a reward for beating the game.

>>29946932
That is the post-game. Like >>29946823
bar gen 2, I also accept that post game begins after becoming champion.

>>29946810
The main gems after the cha pion, but every game has story elements that come after. Its why the "everything after end credits is non cankn" retards are, well....retards