Yes, my vote would go to the CP50 for jazz playing. It might not be the most complex sound for AP, but it works...
You might want to check out the Yamaha CP33 if you can find one around. Many people still prefer that sample to newer CF Grand sample of the CP5/50. It's the same GH action as the CP50 and even cheaper. Personally I still prefer the newer CP50 over the CP33.
I like the immediacy of the Yamaha sound to action connection much more then the Kawai. The MP6 always felt like it was playing catch up with itself with regard to the sound and the way the action interacts.
It's a taste thing and certainly not all inclusive, but I think you'll generally find more jazz oriented pianists leaning towards Yamahas.
Yes, I had the CP5 for almost 2 years and liked it more then any electronic keyboard I'd ever owned in 44 years of buying this stuff. At 59 though, the schlep became too much for many of the smaller, cramped places I was playing, so I went with the more compact Nord Piano 2...which is excellent for what it is.
We're all hoping that Yamaha previews a new, lighter CP5 type DP at NAMM in less then 2 weeks away. It might be worth it to hold out (if you can) and see if anything new is in the works. However, it might be April/ May or even later till its available in New Zealand.