Are you doing stopovers in all cities? If not which ones are transit only?

Any AMEX Membership Rewards points?

My initial thought is that all Economy is downright brutal for a RTW. Just thinking slightly more outside the box here, if a lot of these cities are transit then I think it could be well worthwhile to include your AA miles and string together a bunch of one way tickets on both Star and Oneworld with the long segments in Business.

You are correct, those Chase points can be transferred to UA easily. But they can also be transferred to BA Avios. Your short intra-EU segments are very cheap on Avios. AMS-CGN on AB, CGN-LHR on BA. SYD-AKL is viable on Avios as well.

At the end of the day you'd probably end up spending more than 200k miles but could be much better rested and far more comfortable on the way.