The go-kart place in western Taichung is good. I wrote about it here;
[Rosso go-karting in Taichung
The owner, Rosso, is pretty safety conscious. If anyone drives like a complete idiot they get pulled over for a chat and if they continue their drive is terminated. That makes it a good place to take a mixed-ability group. You can still get a good competitive drive though.
They only rent out 270cc 4-stroke karts (Bowman), but they’re still faster than most rental karts in western countries, which typically are 150cc.
I think there are about 6 tracks in Taiwan for semi-pro karting. There’s a league and it’s on TV sometimes. I imagine Rosso’s is one of the places and maybe the Longtan one is as well. Don’t know about the others.