Good idea. Maybe I wouldn't limit team to be 6 Pokemon. Some shows have fewer main characters. And sometimes, characters leave the show and new one appears.
If a show has one to two main protagonists, you fill your team with antagonists or side characters. Again using Code Lyoko as an example, William was a secondary character who was recruited by the group but was turned into a tool of evil by the antagonist on his very first mission. Takes the protagonists the rest of the series to free him, all the while he battles them while under the influence of the antagonist(and is a pretty formidable foe too). William was strong and had a bit of a temper like Yumi and therefore Onix kind of matched his personality(that or Mewtwo but that would be cheating).
If Generation 2 was available to me, I'd pick Milly and Tamiya as Plusle and Minum. These two were best friends and were always supporting one another as Journalism enthusiasts. There are several more characters I could pick as well and I limit it to 6 because I try to give myself a challenge. If a person wants to have no limit on the amount of Pokemon they catch, that's fine but I'd still limit Pokemon captures by the character and Pokemon they best represent(which if you do the challenge, you announce right at the start of the challenge). There would be no first encounter restriction, only the restriction of capturing Pokemon that fits that particular character. Run out of characters and well you can't capture Pokemon.
Some shows people could do(and example Pokemon for each one):
TMNT: Blastoise, Torkoal, Samurott, Greninja, Raticate, and Bisharp. First 4 as the 4 Turtles, Splinter as a Rat and Bisharp as the antagonist Shredder. Then Rhyhorn for Rocksteady, Krookodile as Leatherhead and so on.
Gravity Falls: Dipper and Mabel(Male and Female Meowstic), Zoos(Slakoth), Wendy(Alola Vulpix or an Ice type), Ford(Alakazam), Stan(pick one lol).
This mode has numerous applications and replayability because you can do a new show each time and even if you go with the same show multiple times, you can probably pick a different Pokemon for a particular character, thereby giving it a fresh take.