It appears the Koreans did not go far enough / did not try hard enough. The better course of action would have been to replace the incompatible bits of their language with things more manageable with IME-less hangul.
Arabic script, for example, is always cursive script, which requires a computer to link the characters when typed, which is a similar problem that hangul has, where the character blocks are broken down syllabically. You could create a phonetic alphabet keyboard covering every single possibility, but the issue here is that you have over 11,000 possibilities, which in the end puts you in the same predicament that Hanzi and Kanji are in, where you end up with these amusingly difficult to use typewriters.