For a student focusing on North America, I’d have to agree that a semester of political science/American studies/international relations classes at Michigan would make more sense, with Sydney a good place to go backpack or for a WHV.
(I understand the op has been to the US before, but visiting a country and living there is different. It would not matter if the OP were specialized in the Eurozone or the Pacific region, but to be a specialist in Northern America it seems like a nobrainer to me.)
I have a different opinion – for a one semester program I’d follow my heart.
I would expect Michigan to be academically more challenging, and to be more of a “work hard, study hard” sort of experience. By the way, I have had a couple of business trips to Ann Arbor, so it is possible to go there without attending a university! During one business trip for a week in December the temperature never quite got up to 0 degrees F.
I also have had a couple of business trips to Sydney. As a place to visit it is very nice, and is very different from Ann Arbor. I think that academically it would be very good, but it would probably be a more “fun” experience than Michigan.
Also, if you want to study international relations, note that it is actually a bit difficult to understand the world from inside the most powerful country in the world. Everything is slanted by the distortion of being in such a powerful country. This does not necessarily say which university to attend: If you spend a semester in the US you might be able to see some hint at the “US is all powerful” distortion, which might be a valuable thing to get a hint of. If you spend a semester in Australia you will get a “sparsely populated not powerful” country view of the world.