In my opinion, the way that Mario Kart 7 adjusted tricking and added gliding and underwater driving was excellent at making it feel more Mario-y without falling back on visual clichés. You're timing jumps and bounces, and adventuring through land, sea, and sky, elements that deeply reflect the original Super Mario Bros.. That parallelism makes it feel less arbitrary and purposeless as a Mario-themed sports game because fundamental features that define the main series are cleverly translated over.
Also gliding was significantly better in its debut. They sucked a lot of the fun out of the controls in 8 where it does feel pretty whatever.
I liked how antigravity added onto this by reflecting the new frontier of Super Mario Galaxy. To me, the gravity-warping gameplay felt like the next evolution of the series and even platformers as a whole, just as much as the jump to 3D. Didn't end up being the case, but still. I'm really split on whether it ought to return. A break from 8's set of mechanics would feel fresh, and that's the headliner. But i think not being so pinned down to the earth imparts a really freeing feeling that's way more valuable than the subtle camerawork and smooth curvatures tend to make it feel. It makes some of the flatter courses seem even more boring and antiquated, and i had the same feelings about the BCP for how much it downplayed antigravity. But that might just be me, and i haven't quite figured out why it hits me so strongly.
Coins can go, i absolutely agree with that. Their stat effect is imperceptible and they make Item Boxes leave you a sitting duck—you know this all already. Unlike the other mechanics, it feels like they were reintroduced just for the Mario iconography. You couldn't even buy anything with them—they just unlock random vehicle parts, which removes the potential player choice and expression—but maybe that's for the best since you can only get 10 per race. Now i'm just imagining a 2D-Mario-style system, where the counter persists between races, and whenever it hits 100, you know you're get something good for longterm play (even as simple as a shop voucher). But ultimately i'd still prefer to go without them for a while.