In terms of the Pokémon lore, it's not that trainers are only using 6 Pokemon cause of some rules, it's that they physically can't use any more, as soon as they catch a 7th Pokémon it is immediately teleported into the PC. It's not even like this is just a gameplay mechanic as this is also shown in the anime.
When koopa was still long of time and super star ran out of time and Mario has stepped on koopa shell and 2 goombas got attacked by koopa shell (Good Ending)