I did almost all of my RTWs so far in F (mostly on Star now, but this is a different story).
For short flights the difference is not really worth the price difference. Inside Europe you will not find F, in the US F is crappy anyhow and you can fly F with a DONE, inside Asia there are a few flights with F but if you don't specifially plan your routing you will end up in flights without F anyhow. Same for South America (LAN does not have F)
So it comes down to F between the continents. Are you willing to pay an extra 1000 $ (your cost difference will likely be somewhat different) for the 10 to 15 hour flight?
My personal opinion is: absolutely on CX, JAL. Perhaps on BA and QF (not been on the QF380 yet). No way on AA. So your routing does not look like an AONE makes a lot of sense.

Your routing is also not ideal for Star. Their Sout America Network is almost non existent, also your only choice between the South Pacific and the US is UA and even their new F is substandard.