You would also have at least 7 segments in N. America, which is one too many.

It looks like you would be arriving in the South Island of NZ (CHC) in midwinter, probably not the best time to see much, unless you want to head for the ski slopes. You might substitute BNE-AKL and head to the north end of the North Island, or maybe the Coromandel Peninsula, for some lovely country and the possibility of better weather.

Also, July/Sept. in the Caribbean/Miami can (will) be hot and potentially rather windy (as in hurricane season) - just a heads up. What about Alaska instead (DFW-ANC?)

Consider HNL-LAX and do a loop trip by car from LA to SF and back (or v.v.) - saves a segment for better things?

Finally if you like "skylines" then Chicago should really be on your list.