But… isn’t Amagasaki literally the English transliteration of 尼崎市? Isn’t a name a word?

It’s a romanization, not specifically an English one. Romanizations, or transliterations in general, don’t translate from one language into another, but from one writing system into another. You’d have more of a point if the word was substantially changed for English, for example like the English Munich is quite different from the original German München.