Public are much more likely to leave you have a natural birth!
Private wants to make money with C-section, that's why they will push you. The other reason is that if you pay a specific Obgyn by advance he will turn your natural birth to a c-section if he finds the labour too long and doesn't wish to wait !!
Public hospital are very good, they let women have a long labour without rushing them to an unecessary c-section
If you are willing to travel for the birth of your baby and don't want specifically HK, i recommend Singapore.
I will have a water birth with the Pr Chong Yap Seng, at NUH hospital
I left HK 2 months ago at 7 month of pregnancy, there is no need for paying by advance for saving your chance to give birth there. They have great facilities and it's cheaper than HK (specifically as a non resident)
Feel free to PM me for more info
good luck:flower: