It sounds like while there are a lot of reused components, there's a lot of tweaking needed for it to look right:
> For instance, just have a look at the various uses of the radical 言 (yan, speech). Even in cases where it is in the same position, such as the left half of the character, the stroke weights and shapes are slightly different (the characters shown below are the exact same font and size).
As for input: there are component-based input methods where you would for example type 明 by entering the keys corresponding to 日 and 月, but as far as I know, they aren't very popular. Most people use phonetic input systems where you type e.g. 明 by typing "ming" and picking it from a list (or letting auto-correct do it for you).
Upper line - complex kanji, Lower line - its components.
When you look at the complex kanji on the upper line, some of its components look unnaturally stretched, as if they were automatically scaled instead of being drawn by hand. So I'm guessing the design is at least semi-automated.
(It's common for kanji like 日 and 雨 to have slight visual differences when used as components – the tiny 'legs' 日 has are missing in 曇 and 温, at least in the font I have. But it still looks like they just have a few variants of each character that are reused)