Seems especially weird considering there's input methods for just about every language and script on the planet, but none for two of the most commonly spoken languages in Asia… am I missing something, here? When selecting an input method, "Asia" there's about a hundred languages, including Chinese, but no Japanese and no Korean.
Seems like a huge problem to me. Maybe I'm doing something wrong?
A volunteer has actually started working on support for Korean recently, and it may be deployed soon. However, the Korean script is actually quite simple and a keyboard for it is relatively easy to implement.
Japanese, in contrast, is very complicated. In theory, it can be done, but it will require very complex mapping between syllables and more than a thousand characters. Because of the complexity, the loading time can be prohibitive, too.
Chinese indeed has an input method in Universal Language Selector, but it only outputs Pinyin, which is a relatively simple Latin-based transcription system. A full-fledged input method for Chinese characters would be a bit similar to Japanese, but even harder, so it's unlikely to be implemented here.
Luckily, Korean, Japanese, and Chinese don't actually need special support, because they have been well-supported in all major desktop and mobile operating systems for more than twenty years, and the people who write in them know how to enable and use these keyboards.
The keyboards in Universal Language Selector are mostly targeted at people whose languages are either not supported in operating systems at all (this is true for many languages of Africa), or languages that are supported, but aren't actually used very much by the public for various reasons (languages of India).