The section of the rulebook quoted is the pertinent one. There aren't any "forbidden" bits in the hunters in the US. You can see that there are recommended bits - these are the bits that are considered conventional and will not be penalized. You can technically show in the hunters in any bit, BUT, as the rulebook states, the judge can (and, in practice, always will) penalize for "unconventional" (aka anything that's not a snaffle, pelham, full bridle - but you never see full bridles in the show ring anymore) bits.
All the bits you mentioned that are banned from Show Hunters in NZ would be considered unconventional and penalized in the US hunter ring as well. While it wouldn't be a disqualification, you would never get a ribbon if you went in the US hunter ring in a gag, Baucher, Kimberwicke, etc.