Even if Westerners maintained the characters for the names rather than transliterate them, there's still the problem that common surnames are used by literally hundreds of millions of people of Chinese descent, and there are a lot of commonly-used given names as well. I think in all of Chinese, Korean, and Vietnamese almost all given names are 1-2 syllables in length, so there's even less space for disambiguation, even before you get to the additionally ambiguities introduced by transliteration and dropping of diacritics.
Japanese has several advantages:
1. Both given names and surnames often use more than 2 syllables. This makes it more familiar to speakers of European languages, where multi-syllable words are no big deal.
2. English speakers find Japanese easier to pronounce and remember due to its simpler syllable structure and use of globally more common sounds compared to the other languages.
3. Unlike the other languages, Japanese names are always given in Given Name, Surname order in English media, while names from the other languages are inconsistently transcribed in either order.
4. Japanese is the "default" East Asian culture for the West in many ways, since it's the one that at least the US has most deeply engaged for the longest time. Many, many cultural exports that actually originated in China were introduced to the West via Japan with Japanese names. Also, Japan has been an advanced industrial economy since the late 1800s, while China only achieved that status a few decades ago.
5. Japanese transcription into Roman characters has remained consistent for a long time (helped by the simpler phonology). Chinese and Korean have been cursed with several incompatible transcription systems over the years - although Pinyin has been winning out for Chinese, there are still many names with legacy transcriptions, not to mention the other Chinese languages needing their own transcriptions. Vietnamese does exclusively use Latin characters, but the diacritics and unfamiliar pronunciation makes them harder to remember.