Why is it that John Chrysostom is almost never referred to as "John Golden Mouth" in English?
Solution 1:
Actually, the Greek is Χρυσόστομος, so writing Chrysostom is in some fashion already an Anglicized translation, since we have transliterated Χ (chi) > Ch and have dropped the -ος at the end. Mozart kept the Latinized Johannes Chrysostomus in his own name.
I have another reason, though. The nickname Χρυσόστομος shouldn’t really be translated into English “golden mouth”, or else you’ll lose its true meaning. The culturally correct version for English is actually “silver-tongued”, because that’s the corresponding idiom in English used to refer to one’s eloquence in speaking.
However, sometimes we really do get to read Ἰωάννης ὁ Χρυσόστομος fully translated, as Maggilory did when in 1871 he published his work, John of the Golden Mouth.
I still think John the Silver-Tongued sounds better in English, because you have to translate the sense, not the words.
Solution 2:
St. John Chrysostom is very rarely called "Golden Mouth" in English, because there is no English idiom "golden mouth". It sounds very strange and unintentionally hilarious.
However, there is an English idiom "golden-tongued", and lo and behold St. John Chrysostom is sometimes called St. John the Golden-Tongued.
Solution 3:
Where a name or nickname can be reasonably rendered by English-speakers, generally it's not translated. Chrysostom is easily said, as is Charles André Joseph Marie de Gaulle — we don't render his name as "Charles Andrew French". Even a transliterated name like Aung San Suu Kyi is reasonably easy once you know how to say it.
However a name which can't easily be transliterated like Tȟatȟáŋka Íyotake may well be translated. Most people have heard of Sitting Bull.