It totally is a throwback to the cold war, but here we are living the real world where some nations get their own separate English term for the job of working in space.
The etymology of astronaut is from Greek astron (star) and nautes (sailor), and the assonance with argonaut - a sailor aboard Jason's ship called Argo. It was coined by a Belgian, mirroring the French aeronautique. Cosmonaut is from Greek kosmos (universe, in Pythogorean usage) and nautes. Each of these words would seem out of place if used in translation - one would almost certainly find it as "Chinese astronaut" or "Chinese cosmonaut" in every actual usage.
I find taikonaut to be a very cool word that blends eastern and western language history in a modern, globalized reality. I like it, even though the Chinese don't use the term.
None of these words are conspicuously English, though. Spacefarer and spaceman probably have the most grounding in English, with etymologies going back to at least Middle English. All of the root words are still quite recognizable in their meaning (unlike astron and nautes). I think it's interesting that the actual Chinese term in use is closest to spaceman, which would probably never be used in translation because of the dismissive, even comical, connotation the word has in English. It's also unlikely that the term would ever be left as taikong ren, because it has no meaning for English speakers. Taikonaut it is.