It’s not something anyone else in the world seems to struggle with, where there are *.gov.uk, *.edu.au etc.

If anything the .gov, .mil and .edu being just American is confusing, as well plainly inappropriate (it feels like an American cultural imperialist thing to people from outside the US). It would have been much better if those had been retired decades ago and moved to under the .us TLD, so e.g. whatever.edu would become whatever.edu.us like every other country. Any existing domains on .gov, .mil, .edu etc. should only be allowed to exist as 301 redirects.