Kinda forgot that Star Spin reflected things. It's been a while since I've played Galaxy, but yeah, having it replace cape makes more sense now. As for Down Air, it's honestly one of the most unique ones in the game so replacing it with a generic stomp or divekick doesn't appeal to me even if it means having two spins.
Not sure how you missed this but I'll say it again. In no canon iteration have I seen Mario have any kind of stomp. He has the ability to land on things and hurt them but that's not the same thing as a dedicated action. He doesn't thrust his feet downwards in an attack or anything. You're making a move out of nothing.
Greninja has a ninja dive kick. Mario just bounces off heads. That's what the difference between a stomp and a footstool is. One is an action, the other is just making use of a situation.
Mario excels at creating that situation but all he's actually doing is abusing his own portliness.
Giving him more air control with the luma spin, on the other hand, definitely appeals to me. That plays right into Mario being Mario while drawing directly from one of his games. It is, however, already one of Rosalina's moves, even if her version works very differently.
That can probably be overlooked, though.
Who has an armoured double jump and why?
Your point on PvP mechanics clearly overlooks the fact that characters' movesets aren't just for 1v1 no assist trophy battles but the rest of what you say rings true. Gross as it may be to contemplate the idea of violating footstooling like that. Where do we draw the line, though? Please tell me Squirtle's Project M turnaround hitbox is still utterly wrong in every respect.
Completely agree with the first part, which is why I'd rather have Goomba Stomp be worked in as a unique footstool.
I don't know why you're so against expanding on one character's footstool so much. The point of listing off things like Kirby's multiple double jumps, Yoshi's double jump armor, or Yoshi's unpokeable egg shield is that some characters already deviate from the norm when it comes to certain universal mechanics. Some other ones I left out include Peach's float, Ryu's funky air movement that makes him feel more like Street Fighter, Bowser's light armor, Link's shield, Bowser Jr.'s damage calculation which differs between his body and car, DK's Cargo Throw, Bayonetta's Bat Within dodges, Corrin's charging hitbox on FSmash/Ness's charging hitbox on Yoyo smashes in earlier games, wall jumps/clings, tether recoveries/Zairs, crawling, and even double jump canceling in past titles. If there are characters that don't stick to the generic template of these universal mechanics, why is slightly altering how footstooling works for one character "violating" it? The point is that not every character sticks to two bland jumps, a normal shield/dodges, and everything else I mentioned or didn't bother putting in. Character unique properties have always been a thing, so there is literally no reason to argue against this unless you just don't like the idea to begin with.
I don't have a complete idea of what a buffed footstool would be like, but I'd imagine it would be some combination of doing damage (only when you get the footstool. This does not mean having a hitbox on a jump unless you don't know the meaning of the word. It's more akin to the passive damage Pichu takes from using electric moves than anything.), sending opponents downward faster, giving Mario more control over his jump height, or putting grounded opponents into a missed tech state/longer stun time. Giving a small nod to footstooling's inspiration isn't all that much to ask for. And yes, there are more game modes than just traditional FFAs, but the inherent weakness of a Goomba should not take precedence over the most important aspect of the game when the two things aren't even mutually exclusive.