I do agree Asia Miles is quite stingy on reward availability, whereas I can easily find something, with connections, with my UA/AC points. My experience with Asia Miles has been to wait for last-minute availability, checking several times a day starting from 1 month before departure. The waitlist doesn't seem to clear immediately when a reward seat pops up for some reason. Some forumers here have speculated the waitlist clearance happens at set times rather than real-time.

I have an impression Star has far more reward seats available than Asia Miles.

You have 2 difficult segments to find seats. AKL-HKG is only 1 flight a day and HKG-FCO is not even daily. You might need some oneworld partners in your booking to make it work, likely with an extra stop in LHR.