I wouldn't say that his role in those games was "major", but that's really up to personal opinion.
Well it is your own take on it. However, you must agree that it is pretty hard to not notice Toad's presence in these games especially.
Also, yes, the amount of spots available in this game is important to seeing whether Toad can get in.
To keep going with my list (sorry I just need to note his extra appearances since some requested it):
Mario RPG:
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Mario RPG: Legend of the Seven Stars - Toad himself was a pretty important story mode character in this game. He appeared in the beginning and helped you during this part. He then made many cameos and even ran a shop in Bowser's keep. A pretty big role for a NPC character.
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Mario & Luigi: Superstar Saga - Toad was actually playable in this game. Who knows why, but he did receive a playable role in the intro and this is pretty noteworthy.
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Mario & Luigi: Partners in Time - Toad appears to be the Toad who is with Mario and the gang during the events of the time machine and etc. While all the other Toads flee, Toad himself appears to stay with the group. He compliments on Luigi's jumps and beside Prof. E. Gadd for the rest of the game. He even appears in the ending during the time when Bowser swallows the shroob mushroom and he also appears in many of the cutscenes (check the ending one; has one Toad only). While his role is minor; it is nice to see Toad himself referenced like this in front of all the other Toads in the game. Also for the Toads, the capturing of their VIM was essential to the storyline; so, the Toads were important as a whole.
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Super Paper Mario - a very definite and confirmed appearance of Toad himself. While Toads appeared in all the Paper Mario games as NPCS, Toad makes an appearance in the intro. He even has his own catch card. Also I think there is a mini-game with Toads there as well.
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Satella-Q - a very unknown game that was released on the satellaview for the Famicon. Apparently, Toad is the main character in this game (another starring role).
Extra:
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Fortune/Itadaki Street Wii and DS series - cross-over between Mario and the Dragon Quest series. Interestingly, Toad appears in both games as a playable character (representing the Mario series).
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Yakuman DS - a game in Japan only and Toad is one of the playable characters.
Roles of his species that I did not mention:
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Paper Mario 3DS - While Toads appeared in every PM game, we will finally be getting a Toad as a partner in this game.
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Mario & Luigi: Bowser's Inside Story - The Blorbs were important to the story line (disease affecting the Toads), and the Toads made common appearances. Also I found it interesting that there was a Toad to represent the mic option in the main menu (why not any other character?)
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Mario vs Donkey Kong series - Toads were everywhere in this game aside from the playable Mini-Toads. Toads were pretty much the only Mario series characters in this game aside from Mario, Pauline, and DK (as well as the enemies) to appear in all of these installments.
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Mario Strikers series - Toads are playable characters in this game.
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New Super Mario Bros. DS - Appeared in the mini-games and a minor appearance in the intro.
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New Super Mario Bros. Mii - I forgot to mention this before, but Toads are playable in this game, and seeing the development for this game, I wouldn't be surprised if it ends up being an actual game.
Notable Cameos of Toads:
Kirby Superstar Saga - Toad was apart of the audience in both the original and the remake games in the dedede stadium.
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Chase Mii Wii U - Miis were dressed up in either Mario or Toad costumes. Notable seeing how only Mario and Toad were the only characters to use as costumes for your Miis.
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Animal Crossing series - Toad has his hat available as an accessory and a t-shirt called Toad shirt.
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3DS Mii channel - there is a Toad hat.
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Nintendo Monopoly (both) and Mario chess - Not video games, but Toad is present as a Mario series representative in all of these board games.
I don't know if I missed anything, but if I did any of you can correct me. I apologize if you found this useless, but I posted all this so people know that there is a thing known as "THE Toad" and what relevant roles he plays. Also the roles of his species.