Also, I wonder which "special qualities" Waluigi has but not Toad, besides some differences in stats. It seems that all the Mario cast is more or less on the same level in the sport and party games.
Wind-based and Water-based attributes (which Toad has neither of) in addition to the far greater skill in jumping.
That's more of a conflict with game design and it's not even remotely true.
In Mario Party, you usually don't have that control of what minigames or events you play amd even in the modes when you do, it's still a casual party game where it makes more sense for everyone to be on an equal footing. Generally, the minigames you play will be decided on a number of random elements. Keeping all the characters the same keeps things more skill reliant in an already luck reliant game. For example, making certain characters faster at mashing would favor them more in minigames where pressing the A button is all that's needed and depending on how many minigames do that,it just creates even further imbalance.
It's like trying to say Pokemon's roster is balanced just because their power gap is almost non-existent in games like Trozei. Anyone that plays Pokemon on a high level knows how laughably unbalanced Pokemon is as far as monster collectors go.
In sports games, where it is established that he has greater qualities in technique and defense, you're going to have more control over what events and minigames you enter, so the characters you pick for what events actually matter no differently from how they would in actual sporting events where the best of the best qualify to get their representation.
It's no different from how we have complete control over how Smash is played to any preference and then choosing characters depending on the format.
Characters that can get kills more reliably amidst a ton of distractions like Ganondorf, King Dedede, King K. Rool, etc are going to have greater favor in timed free for alls as opposed to faster, lighter characters like Pikachu whom excel in 1v1 stock format.