> How would you write, in Chinese, the words thingamajibber, gizmosity, or half the things that come out of AvE's mouth?
With Chinese characters, of course. Why wouldn’t you be able to?
In English “thing”, “a”, and “ma” are already words, and “jibber” would presumably be the first character in “gibberish”. So you could write that made-up word by combining those four characters.
> But how do you write it down without a phonetic alphabet?
In general to write a newly coined word you would repurpose characters that sound the same as the newly coined word.
Every syllable that can possibly be uttered according to mandarin phonology is represented by some character (usually many), so this is always possible.
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Regardless, to reiterate the original point: I'm not claiming Chinese characters are better or more flexible than alphabetic writing. They're not. I'm simply claiming that there's no inherent property of Japanese that makes it more amenable to representation with Chinese characters than English is (other than the fact that a lot of its vocabulary comes from Chinese, but that's not a real counterpoint given that there is lots of native, non-Chinese-derived vocabulary that's still written with kanji).
It would be possible to write Japanese entirely in the Latin alphabet, or English entirely with some system similar to Chinese characters, with minimal to no change to the structure of the language.