Mario Kart is leagues and leagues beyond any and all these other sport spinoffs just by sheer sales numbers and Toad's the second most popular Kart racer according to statistics right after Yoshi.
They cut Toad, you're damn ****ing right there will be outcry. He's become a staple. When he looked like he wasn't in Mario Kart Double Dash there was outcry about it, only for relief when he turned up as a secret character to introduce his new partner Toadette.
Mario Sports Mix Toad brought every single one of his cousins along with him...which is the only sports mix game that was made. If you're on about Superstar Sports...which is one of the worst sports games ever recieved from the Mario crew...kinda glad he wasn't in that honestly.
Mario Party's a weird case since him and DK swap who gets to be the party host often. In that game not being playable means he's the most important character usually in deciding who wins the damn game, so I'm not sure if I can count that as a minus.
But hey, you're not counting Mario puzzle games for some reason so I guess I don't really need to count it as a minus that Toad doesn't appear as a playable character in Mario and Sonic since it's half a Sonic game. It's as arbitary a reason as 'a simple puzzle game' after all.
Waluigi as far as I've seen has only ever been popular amongst Smash fans. And that's mostly to fulfill the Mario and Luigi vs. Wario and Waluigi.
Geno is less important than Waluigi to Mario as a whole but Smash fans want him more than him. Being more wanted doesn't make you more important to the franchise, else Geno's inarguably the sixth most popular Mario character after Mario, Luigi, Peach, Bowser and Wario.
As for Smash support numbers; they dwindled after Toad didn't turn up in Brawl. Due to the Toad rumour in Melee and being a popular choice from 64 a lot of people assumed he'd make it into Brawl. When it became clear Sakurai wasn't going to shift, Toad fans began accepting it wasn't going to happen. He still had a decent amount of support in Sm4sh rivalling Bowser Jr for the top dog Mario spot. Captain Toad bolstered him a bit too, and he's usually in the high 30s from what I've seen.
Waluigi's huge push seems to have come mostly between Sm4sh's base roster and the DLC ballot. Now Daisy, who was basically Waluigi's female spinoff rival, made the Smash cut, the push for Waa has grown.
If Luigi doesn't matter then how on earth does Waluigi matter?
Not that it matters because unlike Waluigi, we can see if Luigi and Toad play differently in the mainline games(and sometimes they do). Honestly if Waluigi is ever lucky enough to make it into the mainline Mario games I doubt he'll be using any of these crazy abilities you seem to think he has.
Source on that backstory, because I've never heard it before. Sounds like bull**** because barely any character in Mario has a canon one, and I doubt they'd give one to Waluigi of all characters as a result. Captain Toad has a story of becoming his own character anyway thanks to Sunshine, Galaxy/Galaxy 2, Treasure Tracker and 3D World/Mario Odyssey.
Captain Toad is full of personality, a single video of the gameplay showcases that.
Toad meanwhile also has plenty of personality; Super Mario RPG, all the interactions with Wario and Luigi in Mario 64 DS, Mario and Rabbids, Mario Tennis Aces...I'm not convinced you've actually played many Mario games and have just decided to hate Toad.
Abilities huh?
Waluigi has a bunch of random one off moves that he doesn't use across games. If we're counting the ability to spawn whips and swimming through air as Waluigi's powers and abilities we might as well include Toad's ability to lift 8 times his weight, manipulate the environment around him by spawning mushrooms and walking up walls since they have as many appearences, if not more.
Waluigi's only consistent abilities is he sometimes uses bombs and he's pretty good at tennis. Toad's consistant abilities are his super strength and speed.