Made a quick image about how softdrifting works so that people finally stop getting confused.

The orange zones are where the stick has to be to get the fastest mini-turbo charging rate.
In the white zone, the mini-turbo will charge "normally", as if the stick was left in neutral position (gray).
The red lines are roughly the ideal positions for optimal softdrifting (getting the fastest charge while turning as less as possible)
The majority of people softdrift by tilting the stick upward while turning, but as you can guess from the image above, pulling the stick in a direction but not completely all the way also works.
Notice how the 45° angles are a bit off the red line into the orange zones. This means using a D-pad, or the octogonal indentations on a Nunchuk/Wii Classic controller, will allow you to softdrift, but not in an optimal way. While this is good enough for most courses, this can be troublesome in some places like the frequently mentioned first turn on MKS.
Ultimately, the controllers with the round ridges (Gamepad and Pro Controller) are the most convenient to use (imo), since it's easier to fine-tune the level of drifting during a turn with them, and stay on the best line, but you only ripe those small benefits at high level play, so stick with what you're the more comfortable with unless you want to reach the top.