(If there had been the kind of rendering technology that is used for Indic text today back when Korean text processing on computers started, chances are that the syllable grouping would be handled as rendering-time shaping and not as an input-time IME issue.)
Additionally, Korean IMEs have a feature to convert a word into Hanja, but it's something you need to take action to invoke as opposed to Japanese IMEs offering to convert to Kanji by default.
...that's the opposite of the comments that triggered my question. Multiple people said space is needed to trigger it, then backspace to remove an erroneously-added space character.
Is the answer that they're using it wrong, and are actually inputting a space directly because the IME already acted for them?
On a laptop, my experience is that romaji -> hiragana is as-you-type due to being unambiguous, while the default trigger for hiragana -> kanji is the spacebar, same as described for Korean. Hence my confusion as to how it's different - the individual characters certainly are, but it sounds functionally identical.
On my phone just now, typing this comment, the experience was switching to a Japanese keyboard and inputting the hiragana directly, then the kanji suggestions appeared where autocorrect suggestions normally would.