to be honest, the wiimote isn't as uncomfortable as it looks, for a good mariokart experience i would reccomend you just buyng some Nunchucks. and everything should be alright.
you'll get a similar experience as Switch detached joycons. most games don't use as many inputs as what's avaliable on a pro controller. try playing with wiimote/nunchuck first, and then if you really feel it won't work, search for pro controllers. since they can get expensive, i'd put that as an after purchase.

Wii and wiiU use the exact same controller, So that technically is a non issue, you can still use the WiiU Gamepad for all controller related games.

In terms of price, there won't be a huge difference between a WiiU or a Wii.

Games, WiiU being able to play wii games also is a bonus. There's hella tons of chill games to play, MarioKart8 appeared on wiiU, you could still play MKwii, Zelda Breath of the wild started there too, so it's good.

Controllers: Don't overthink too much about that. Only a small 10% of the games on both make a real use of 3rd party controllers, the VAST majority of them use for the wii, Wiimote+(sometimes)Nunchuck, And for wiiU most of them are meant to be played with the bug Gamepad with the screen.

Battery concerns. WiiU and therefore Wii's Wiimote (the stick looking controller) use 2xAA batteries. Not rechargeable, so no issues in terms of battery life with usage.

On the WiiU's gamepad tho, there is a rechargeable Li-ion battery inside, that can indeed be really be used and don't last as long. Plus side, the battery is accessible from the back of the gamepad, 3-4 screws to remove, pop it out and drop a new one for around 10 bucks.

In terms of emulation and maybe hacking the console to not buy games. I would highly reccomend using the WiiU as it can play Wii games natively with the hdmi out port, so CLEAN AND BEAUTIFUL 720p display. Against the wii 480i composite cables Homebrewing the wiiu is really simple, almost as simple as the wii classic.

Downside. There is no Gamecube ports on the WiiU so you'd need a gamecube controller to plug in your Wiimote to play, or use the WiiUGamepad.

If you still wanna get a wii, and see a good deal on a wii mini. Don't. System can't be updated, bcause no Sdcard slot, no useable usb slot, no network connection. (And ofc no homebrew possible)

Tldr:All in all, i'd highly reccoment a WiiU if you are hesitating between Wii classic against WiiU Bcause: -can play both catalogue of games. -controllers are identical -clean 720p hdmi output -easy homebrew

Oh, that's not what i heard, (haven't fone it myself, a friend, and it was a pain in the butt, had to redo and reinstall backup 4 times, even tho we were following instructions precisely)

Whereas my 3ds, done in 30 mins first try...

In terms of performances, and possibilitoes, the vita can offer better emulating, and get to newer consoles than the 3ds, (eg, PSOne)

In terms of how easy it is to install and actually make it do emulation, the 3ds is the easiest of them,

The vita didn't sell well in the western world, therefore not much people developped emulating solution on it, and therefore being really difficult to install.