It won't be easy to capture the feeling of DKR and resist the urge to create a Mario Kart clone. It even happened to "old" Rare: Mickey's Speedway USA was a decent fun racer but ignored everything that made DKR great while copying a lot of Mario Kart's characteristics.
But yeah, in theory I'm all in for a spiritual successor to DKR! My wish list:
- explorable hub world
- collectibles to unlock new courses
- selectable vehicles
- actual VEHICLES (no animals)
- fixed and upgradable items > item roulette
- special races where you need to collect items and win (silver/play coin challenge)
- challenging boss races
- new characters that'll get their own games later
- secret characters
- a consistent theme for each cup
Add some new twists and ideas and there you go!

I agree that "missions" would be a nice addition to a game like this.
DKR was always a lot more about skill than Mario Kart. The single player mode was quite challenging and you just couldn't win with dumb luck - you had to know the courses, which items to use, which character to pick and how to use a combination of sliding and braking to take tight turns.
I have to agree that this is also a disadvantage of DKR... In Mario Kart everyone can pick up the controller and play since there are many items and random circumstances to help lesser talented players. It's not enough to beat some high scoring Japanese pro gamer you meet online but with a little luck they might reach the podium after a few races. In DKR's multiplayer people usually lost their interest pretty quickly when they realized they couldn't beat me due to my single player experience... That's why I was excited to check out the online mode of the NDS version - unfortunately I didn't meet a single player because the online community was pretty much dead when I finally managed to set up the Nintendo Wi-Fi connection a few years ago.
tl;dr: I'd love it to have an online mode since in a true DKR successor you probably won't have a lot of fun in local multiplayer due to the possible lack of skilled combatants.
