I believe that the car was redesigned for Broadway so that it could travel. The show is currently out on a very successful tour here in the UK, so I'm sure it's only a matter of time (unless the muted reception afforded to Doctor Dolittle has put people off touring this kind of show).
Kiddy-Widdie-Winkies, as well as Bombi Samba were written for the stage show
When people ask why the Sherman Bros. didn't write the new songs for Mary Poppins, I always direct them towards the extra songs they provided for Chitty...