If you want to maximize the number of F flights a few ideas (this is from memory when I was planning my last AONEx:
Inside Asia HKG-Tokyo (HND and NRT) have F, SIN-HKG on soem days has F.

Between Asia and SWP I don't think there is currently anything except the BA flight SYD-SIN. Btw BA F in my opinion is just like a good business class.

Between Asia and NA both CX and JL have a very good F product. If you want to go to HNL I believe you need to come from Asia and cannot backtrack from NA - JL but no F.

Between NA and Europe you have the choice between AA and BA. I don't like both products, but BA is still better.

If you want to maximize the F experience you should also consider a Star RTW. It is mileage based and you have more choices for F flights. TG and SQ between SWP and Asia, many airlines between Asia and Europe. NH, OZ and UA (avoid) between Asia and NA.